Washington (CNN) The political network funded by Charles and David Koch is seizing the break on Capitol Hill to try to torpedo a key part of the Republican tax bill, exposing yet another fissure within the GOP and gambling that it won't sink the entire effort.

After months of private hand-to-hand combat in the halls of Congress, the Koch operation is beginning to bring public pressure to bear on a group of Republicans who are poised to support a border adjustability tax, a central tenet of the plan pushed by House Speaker Paul Ryan. And it is doing so by aiming to turn the same political coalition energized by President Donald Trump onto those Republican leaders, the latest pull in the tension between the Kochs and Trump.

It is a third iteration of that topsy-turvy relationship: After spurning Trump during the presidential election, and then helping to block the administration's marquee legislative goal of repealing Obamacare, the Koch operation is now calling upon Trump's very movement to dump a tax on which Trump himself has waffled.

TV ads target border tax

"America voted for change," says a new television ad from Americans for Prosperity that will begin airing Monday, promising that the costs of gas and groceries will increase. "Tell Congress, that's not the change we're asking for."

The campaign, which will spend hundreds of thousands of dollars on national cable television, is an attempt to stir public outrage against part of a bill before it picks up speed in Washington. It is timed with the congressional recess, at a time when the Koch operation sees Republicans as particularly attuned to their home districts and states.

But the new push also carries considerable risk: The provision is only one part of comprehensive tax reform, a priority of the Koch network, and some Republicans worry the costly fight could doom the common goal of lowering tax rates. And it once again positions the Koch operation uneasily alongside the administration, with which the Kochs have labored to mend fences.

Trump himself has see-sawed on the border tax, which would levy a corporate tax on imports to the United States while withdrawing the fees from US exports -- a bid to raise revenue while encouraging the purchase of domestic products. It is seen as attractive to Trump given his skepticism toward large free trade deals and his promise to revitalize American companies.

But Trump has yet to signal that he will spend political capital to fight for the provision, and several of his economic advisers have suggested they personally oppose it.

"We're very much set on the concept of the tax," Trump told The New York Times last week when asked for his opinion of it. But he then added that he didn't want to announce anything before seeing whether savings could be created from a new health care bill.

Opponents of the move, such as the Kochs, claim the hike would be passed on directly to consumers and raise prices of everyday goods. Supporters of the provision, including much of the American manufacturing industry and House Ways and Means Chairman Kevin Brady, argue the US dollar would increase in value, lowering the cost of imports and keeping consumer costs steady.

Tax's backers are ready for a fight

The tax's proponents are leaning into the fight, portraying the Kochs as comfortable with the status quo and warning they would be responsible for comprehensive reform's failure.

"Supporters of the blueprint understand the importance of reforming our tax code in a way that incentivizes investment in the United States and the production of American goods, in American towns, by American workers," said John Gentzel, a spokesman for a consortium of businesses lobbying for the provision. "Meanwhile opponents of the blueprint are trying to stop tax reform and keep the current broken tax system that puts American workers at a disadvantage."

The Koch lobbying operation has conceded that Brady's tax-writing committee is "pretty much dug in at this point," said Americans for Prosperity lobbying chief Chrissy Harbin. But the network is zeroing in on the Senate as particularly fertile ground -- partly due to the number of Republicans who represent large domestic retailers -- and has not conceded the House Republican conference despite the substantial pressure from Ryan and Brady.

"Recess is always a time for resetting," Harbin said. She said she and the network's about 10 lobbyists see the tax as not yet properly understood on Capitol Hill and as therefore susceptible to public pressure. "People conflate it with things like building the wall. And it has absolutely nothing to do with it."

So even as their hefty political operation begins to mobilize in states that their new report says will bear the brunt of the cost increases, their muscles remain trained on the inside game.

Koch Industries spent almost $10 million on lobbying last year, according to disclosures, but the network is not currently lobbying Democrats on the border tax, despite their more protectionist impulses. And perhaps more surprisingly, Harbin said Americans for Prosperity was not lobbying Trump directly -- only as part of an "ongoing conversation" -- even though the White House's official position is uncertain.

"They'll have a position one day and they'll seem to have another one on the other," Harbin said. "The White House is well aware of our opposition."