The Republicans lost the struggle over the Dodd-Frank law to impose new rules on the financial sector, but three years later they’re still tenaciously fighting what’s probably the best-known Dodd-Frank creation: the Consumer Financial Protection Bureau (CFPB).

Richard Cordray, the man President Barack Obama re-nominated to lead the CFPB, had his confirmation hearing Tuesday before the Senate Banking Committee – and what was noteworthy was the warmth and respect that Republican senators displayed toward Cordray.

Sen. Tom Coburn, R-Okla., told him, “I will compliment you. I think you’ve done a wonderful job so far."

Sen. Elizabeth Warren, D-Mass., expresses her displeasure over the apparent holdup of the confirmation of Richard Cordray as director of the Consumer Financial Protection Bureau.

It’s not Cordray they have a problem with – it’s the agency’s autonomy.

Until Obama agrees to changes in the CFPB structure to make it a multi-member board akin to the Securities and Exchange Commission and other regulatory agencies, Republicans said they will block a confirmation vote on Cordray. They also want to subject the CFPB to annual appropriations by Congress. Right now the law allows CFPB to draw its funds directly from the Federal Reserve.

CFPB now enjoys “complete autonomy; the Federal Reserve has no ability to influence their decision making, no oversight capacity, nothing. The sole function of the Federal Reserve is to write a check” to pay for CFPB operations, said Sen. Mike Crapo of Idaho, the top Republican on the Banking Committee. The CFPB, Crapo said, needs to be “subject some kind of accountability or oversight – but in this case there is none.”

Cordray’s response to that argument is that he has testified many times before the Banking Committee and that his agency submits semi-annual reports to Congress.

And Cordray champion and fellow Ohioan, Democratic Sen. Sherrod Brown said at the hearing, “Some (Republicans) here want to nullify the (Dodd-Frank) legislation” and that for the first time in history “senators are blocking the nominee because they simply don’t like the agency that he will lead.”

T.J. Kirkpatrick / Getty Images Richard Cordray nominee for director of the Consumer Financial Protection Bureau, and Mary Jo White, nominee for chair of the U.S. Securities and Exchange Commission, are sworn in before testifying at a confirmation hearing before the Senate Committee on Banking, Housing and Urban Affairs on March 12, 2013 in Washington, DC.

Personally, Cordray sent all the signals of a reasonable and accommodating man in his testimony. And Republicans reciprocated. Crapo told Cordray, “I appreciate our private conversations about the importance of accountability and oversight. I recognize that you can’t say what the White House and Congress will ultimately decide with regard to the issues with regard to… changes in the structure of the agency.” But the Idaho Republican told Cordray he “seeks your support” in resolving the dispute with Obama.

Another Banking Committee Republican, Sen. Bob Corker of Tennessee, told Cordray he appreciated the way Cordray had dealt with him and his staff “and I do hope that over the course of the next short period of time we’re able to figure out a way for the entity to function in a manner which makes everyone on both sides of the aisle feel comfortable.”

But Corker said after he left the hearing that “we’re not there yet” – in other words Republicans still haven’t been able to strike a deal with the White House on redesigning the CFPB.

Dissenting from Sen. Bob Menendez, D-N.J. who during the hearing denounced Republican opposition to the CFPB as “ideological,” Corker said, “This is not some sort of ideological deal. That’s not it at all. People would like to see a structure that’s standard for rule-making – the Fed (Federal Reserve) has it, the SEC has it, the FDIC has it.” Corker said he hoped that before Cordray’s nomination came to the Senate floor for a vote that “there’ll be some breakthrough.”

Asked after the hearing whether he’d seen signs of potential compromise from the White House, Crapo said, “not yet, but I’m still hopeful we can find a pathway. … I do believe that there’s plenty of room for us to find a common ground.”

Meanwhile, a legal challenge to the CFPB is under way in the federal district court in Washington – so even if Obama and GOP senators strike a deal the agency might ultimately be ruled invalid.

In the face of GOP opposition to the CFPB, Obama gave Cordray a recess appointment last year. But Cordray’s status seemed on shaky ground after a ruling last month from the U.S. Court of Appeals for the District of Columbia Circuit which held that Obama had acted unconstitutionally by giving a recess appointment to three members of the National Labor Relations Board on Jan. 4, 2012.

The court held that the Senate was not in recess on that day so Obama could not make any recess appointments. (In the NLRB case, the Obama administration said Tuesday it will file a petition with the U.S. Supreme Court seeking review of the appeals court decision.)

Obama gave Corday his recess appointment on the same day he made the now-invalid NLRB recess appointments.

Although Cordray’s appointment was not directly at issue in the appeals court decision, CFRB foes such as C. Boyden Gray, the former White House Counsel to President George H.W. Bush and lead counsel for plaintiffs challenging the CFPB’s structure, argue that Cordray’s actions are under a cloud – and Senate confirmation of Cordray wouldn’t dispel that cloud. The plaintiffs are the State National Bank of Big Spring, Texas, a small community bank, as well as two conservative advocacy groups, the Competitive Enterprise Institute, and the 60 Plus Association.

“Confirmation of Cordray would not retroactively validate the CFPB's actions taken during his appointment,” Gray said Tuesday. “Those actions remain under a shadow of grave unconstitutional doubt in light of the D.C. Circuit's recent decision in the NLRB recess appointment case, and that shadow will remain so long as the Supreme Court does not overturn the D.C. Circuit's decision.”

Each side has what seem to be plausible arguments: Cordray’s supporters say that Republicans are abusing the Senate’s unlimited debate rule by using a filibuster threat to block a confirmation vote on Cordray.

But Republicans – and the three judges of the appeals court -- say Obama abused the recess power when he appointed Corday last year.