(CNN) Ohio Gov. John Kasich was among the big winners when Republican Troy Balderson apparently survived last week's special election for a US House seat centered on Columbus. Kasich was also among the biggest losers.

In the process, he encapsulated the puzzle facing the small band of Republicans consistently critical of President Donald Trump. Call it the Kasich dilemma. Trump's Republican critics almost uniformly agree that they cannot attract a large audience for their case against the President until Republican elected officials and voters see evidence that his approach is endangering the party's electoral prospects. And yet Trump's critics inside the party feel compelled to help other Republicans win tough races -- as Kasich did by endorsing and filming television ads for Balderson -- in part because they fear that failing to provide assistance will brand them as disloyal and undercut their credibility inside the party.

As a result, Trump critics such as Kasich are left in the paradoxical position of actively working to prevent Republican losses in contests like last week's Ohio special election that would directly strengthen their case against Trump. "It's a good illustration of the conundrum we're in," says Peter Wehner, former director of strategic initiatives in the George W. Bush White House and a leading conservative critic of Trump.

To underscore the contradiction, Trump on Monday used the narrow result in the Ohio special election -- Balderson leads Democrat Danny O'Connor by about 1,514 votes as the final ballots are being tallied -- to attack Kasich. "The very unpopular Governor of Ohio (and failed presidential candidate) @JohnKasich hurt Troy Balderson's recent win by tamping down enthusiasm for an otherwise great candidate," the President tweeted

The very unpopular Governor of Ohio (and failed presidential candidate) @JohnKasich hurt Troy Balderson's recent win by tamping down enthusiasm for an otherwise great candidate. Even Kasich's Lt. Governor lost Gov. race because of his unpopularity. Credit to Troy on the BIG WIN!

Whose support helped more?

The President's argument belies the evidence from the results. Observers in both parties agree that Kasich's highly visible support -- in a district he represented himself under different boundaries during the 1980s and 1990s -- likely helped Balderson hold some white-collar voters disenchanted from Trump in the outer suburbs of Columbus. Given the fact that turnout in Trump friendly rural and small town parts of the district remained modest last Tuesday, it's possible that Kasich contributed more to Balderson's edge over O'Connor than the President's appearance in the district just before the vote.

None of that stopped Trump last week from claiming the result demonstrated that he is strengthening the party. "As long as I campaign and/or support Senate and House candidates (within reason), they will win!" he tweeted . "I LOVE the people, & they certainly seem to like the job I'm doing."

As long as I campaign and/or support Senate and House candidates (within reason), they will win! I LOVE the people, & they certainly seem to like the job I'm doing. If I find the time, in between China, Iran, the Economy and much more, which I must, we will have a giant Red Wave! — Donald J. Trump (@realDonaldTrump) August 8, 2018

Despite endorsing Balderson, Kasich has made the opposite argument, presenting the Ohio close call as a warning signal to the GOP. "It wasn't a good night because this is a district that you should be winning by, you know, overwhelming numbers," Kasich said Sunday on Meet the Press. "So, what you had is, I think, a message from the voters to the Republicans that you've got to stop the chaos and you've got to get more in tune. And stop alienating people and try to figure out how do families do better."

How GOP losses could help Kasich

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Yet, even Kasich sympathizers acknowledge that a Balderson loss in such a previously reliable Republican seat would surely have provided a vastly stronger tailwind for Kasich's case. Which means Kasich played a pivotal role in securing a victory that diluted his argument against the President. "Kasich is kind of caught because on the one hand he wants to be anti-Trump and on the other hand he wants to still be a Republican loyalist, and he's the titular head of the Republican Party in Ohio," said long-time GOP strategist Mike Murphy, another internal Trump critic. "That's how you screw yourself into corners."

Balderson had unique ties to Kasich that helped explain the governor's decision: he had endorsed Kasich during his initial race for governor and his presidential bid in 2016, and backed most of the governor's agenda during his tenure as a state senator. "In this case [Kasich] had a personal relationship with Balderson that dated a decade," John Weaver, Kasich's top political strategist, said in an interview. "He's seen him up close and after hearing from Troy he made a decision. It's not complicated. If it was my decision, it might be something different ... but that's why people get to make these calls."

Still, Balderson provided Kasich plenty of excuses to hold his distance by so closely embracing Trump in ways that directly collide with the governor's positions. On his campaign website, Balderson takes credit for opposing Kasich's expansion of Medicaid in the state under the Affordable Care Act; promises to work for full repeal of the ACA (which Kasich opposed as forcefully as any Republican); and says he "will fight alongside President Trump to deport those here illegally" while Kasich has consistently backed legal status for undocumented immigrants.

Despite those statements, Weaver said, Balderson promised Kasich he would operate as an independent voice in Congress. "Look if he goes to Washington and he becomes a back-benching vote for the leadership and against the interests of Ohio and votes with the president on every issue regardless than he's going to have a real problem with the governor," Weaver said.

Helping other Trump critics in the GOP

This fall, Weaver said, Kasich intends to campaign for other "courageous" House Republicans who have displayed independence from Trump such as Carlos Curbelo in Florida and Will Hurd in Texas. "He's not going to turn his back on the handful of people who have stood with him through thick and thin on important issues," Weaver said.

That prospect highlights the tensions facing the Trump critics inside the GOP. Few contest Weaver's insistence that it would be disloyal for Kasich to abandon the relatively few members who have shown any independence from Trump. Yet by helping those individuals, Kasich is, however modestly, increasing the odds that Republicans maintain control of the House. And that would enormously strengthen Trump both by preserving the majority that now protects him from serious oversight and by allowing him to argue that his direction is still generating electoral benefits for the party -- a point Trump quickly underscored in his tweets about the Ohio race.

The persistence of Trump's support inside the Republican coalition has convinced most of his GOP critics that they cannot open a serious debate about his direction unless and until it threatens the party's hold on power and the advances on conservative priorities, such as tax cuts and judicial confirmations, which have flowed from that.

"There has to be evidence that Trump's approach ... is hurting the Republican Party and specifically hurting Republican lawmakers," says Wehner, now a senior fellow at the conservative Ethics and Public Policy Center. "He's got an iron grip on the party and that's not going to be loosened by moral suasion. It is going to be loosened by rank self-interest."

Murphy, a former top strategist for John McCain and Mitt Romney, sees the same dynamic at work. "I have taken the Leninist view ... that until we have casualties and pain, our 'spark of the proletariat,' there is going to be no grounds for a revolution," Murphy says.

How to oppose Trump and still be a Republican

Yet Trump's conservative critics have been divided over whether they should work to encourage the Democratic gains they consider the indispensable prerequisite to loosening Trump's hold on the GOP.

Wehner publicly says he supports a Democratic takeover in the House because he considers Trump such a threat to the long-term future of both conservatism and the GOP. But he recognizes why other Republicans resist the same conclusion. "You can say that in the abstract but these are very real races, and for a lot of Republicans very complicated races," he says. "People who in any other moment you would have supported, you may not support now ."

But other leading Trump critics inside the GOP, such as Murphy and long-time party strategist Bill Kristol, have not called for electing Democrats. Michael Gerson, the former chief speechwriter for George W. Bush, split the difference in a recent Washington Post column when he urged voters to elect a Democratic House to constrain Trump but to maintain a Republican Senate that could preserve the policy advances conservatives have made under his presidency.

"If Democrats gain control of the House but not the Senate, they will be a check on the President without becoming a threat to his best policies (from a Republican perspective) or able to enact their worst policies," Gerson wrote.

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None of the Republican elected officials occasionally critical of Trump have come anywhere near such sentiments. In Tennessee for instance, retiring Sen. Bob Corker, a frequent Trump antagonist, has been tepid in his support for Rep. Marsha Blackburn, an unflinching Trump loyalist running to succeed him. But Corker has said he intends to vote for her.

Some speculate Kasich feels compelled to support Republicans for the House to maintain his viability as a possible 2020 GOP primary challenger to Trump. But Weaver suggested in the interview that he believes it may already be too late for an internal insurrection against Trump because he has driven out of the party too many of the moderate voters who might welcome an alternative.

"The transformation of the party into a Trump-ian base, at least at the federal level, is near complete," he said.

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That process is bound to accelerate in November, Weaver argues, because the Republican House members most likely to lose are those from swing districts who have displayed relatively more independence from Trump. As the party recedes toward a core more heavily reliant on hardcore Trump loyalists -- both in Congress and its electoral base -- Weaver argues the focus of Republican resistance to Trump may need to shift from a primary challenge toward what he called "something completely different" -- a phrase that suggests an independent presidential candidacy.

"It's already hard to make the case [against Trump] within the party because the party is shrinking before our very eyes and I don't think that trend line is going to stop," he said. "The way to ultimately attack this problem is to make either a clarion call that's attractive (to Republicans) or you do something completely different, and there's time to do that after the mid-term. So we'll see."

In the meantime, though, Kasich has clearly signaled he will continue working to preserve a Republican congressional majority that fortifies the president he appears likely to challenge, one way or another, in 2020. Through November, the Kasich dilemma is likely to only deepen, not just for the governor but for all of Trump's Republican critics