AUSTIN, Tex. — When Texas lawmakers gather here for the start of a 30-day special legislative session on Tuesday morning, they will most likely decide the fate of the Texas version of North Carolina’s bitterly divisive legislation regulating the access of transgender people to public bathrooms.

But something else will be on the line, too: whether moderate Republicans have a role to play in a state party increasingly dominated by far-right Christian conservatives, and whether the last powerful moderate Republican in Texas can keep his job and his influence.

State Representative Joe Straus, the speaker of the Texas House, has long employed a mild-mannered, commerce-focused brand of Republican politics in the mold of former Gov. George W. Bush.

Since 2009, when legislators first elected him speaker, he has been effectively minding the store of Texas. He has kept a relatively low profile and focused on issues like public education while his fellow Republican leaders, including former Gov. Rick Perry, now the energy secretary, and Ted Cruz, the former state solicitor general and now a United States senator, set their sights on higher office and seized the spotlight.

But the debate over whether Texas should pass restrictions on which bathrooms transgender men, women and children can use in schools and government-owned buildings has become a pivotal political moment for Mr. Straus, the presiding officer of the 150-member State House of Representatives. He is in many ways the public face of the opposition to the bathroom bill, angering social conservatives and lawmakers in his party who support the bill and earning admiration from Democrats, business groups and advocates of transgender rights who have denounced it as unnecessary, discriminatory and harmful to the state’s economy and brand.

Mr. Straus, 57, has put himself publicly at odds with the two most powerful Republicans in Texas, Gov. Greg Abbott and Lt. Gov. Dan Patrick, both of whom support the bathroom restrictions. And on the eve of the special session, his ability to hold onto power is under fierce attack.

In San Antonio, Republican leaders in Mr. Straus’s home county, Bexar, passed a resolution in recent days calling for “a change in leadership in the Texas House speakership,” the first time the local Republican Party has given Mr. Straus a vote of no confidence. Other Republicans, including social-conservative activists and Tea Party-backed lawmakers, have threatened to oust him as speaker during the special session.

“His days are numbered as speaker,” said Jared Woodfill, the president of Conservative Republicans of Texas, which has spent a small fortune in recent years to defeat Mr. Straus and his lieutenants in the House. “This is one man who has a liberal agenda for the state of Texas, and who has done everything he can to stop the good work that’s being done by the lieutenant governor and our State Senate and the governor.”

Pro-business, Bush-style, country-club Republicans no longer set the agenda in Texas. What happens in the session — and particularly what form the bathroom bill takes and whether it passes or fails — will provide the clearest signal of whether there is any effective brake left on social-conservative Republicans in Texas.

“In a way now, it’s like Straus and the right wing of the party don’t even speak the same language,” said Brandon Rottinghaus, the author of a new book, “Inside Texas Politics,” and a professor of political science at the University of Houston. “Straus is somebody who is still trying to hold onto the center of American politics and Texas politics. That’s becoming increasingly difficult to do.”

Mr. Straus shows no outward signs of feeling the pressure. In a recent interview in his office on the second floor of the Capitol behind the empty House chamber, Mr. Straus sat on a sofa with a tall glass of ice water and calmly paused for several seconds when asked whether Texas had shifted further to the far right since 2009.

“Well, a lot of the politicians have,” Mr. Straus replied. “I don’t think Texas has. And as the Republican Party continues to dominate all the statewide offices, the competition has been focused, really, on the small-turnout primaries.”

He added, “I’m not embarrassed to say that I know how to govern without being an extremist.”

Mr. Straus said he was not worried about the threats to vote him out after five terms as speaker. “If they want to try to change leadership, it’s in the rules how to do it,” he said. “I’ve been elected five times, so I have to know and be connected with where the members really are.”

The marquee item on the special-session agenda is the bathroom bill. But it is one of 20 pieces of legislation that will be debated, including property-tax reform, teacher pay raises and a ban on abortion coverage by private insurance plans.

The intraparty intrigue heated up a notch on Monday when Mr. Abbott told a conservative policy forum that he planned to publicize a list of lawmakers who support his 20-item agenda and those who oppose it, a list likely to have more political ramifications for Republicans than for Democrats.

Mr. Patrick, the lieutenant governor, effectively forced Mr. Abbott to call a special session, by holding a mundane piece of legislation hostage that must pass to keep a handful of state agencies operating, including the Texas Medical Board, which licenses the state’s doctors. Because the bathroom bill failed to pass during the regular session, Mr. Patrick used that legislation, known as a sunset bill, as leverage to get Mr. Abbott to order the 30-day session.

The North Carolina bathroom bill, passed in March 2016, prompted boycotts by celebrities and led to canceled sporting events and business meetings. The National Basketball Association pulled the All-Star Game from Charlotte, and Gov. Pat McCrory, who signed the bill into law, lost his re-election bid to Roy Cooper, a Democrat, in November.

Opponents of the bathroom bill in Texas predict similar fallout, and those concerns drive much of the moderate-Republican opposition to it.

Few symbolize the disunity among Texas Republicans better than Mr. Straus and Mr. Patrick, whose inability to compromise on the bathroom bill set the stage for a special session that will cost Texas taxpayers at least $1 million.

Mr. Patrick, a former talk-radio host, was the Texas chairman of President Trump’s campaign. Perhaps more than any other Republican, he has come to define post-Perry Texas politics.

Mr. Straus comes from an influential San Antonio family that helped the Republican Party rise to prominence in Texas in the 1980s and 1990s. His mother, Jocelyn Levi Straus, 86, gained a reputation as one of the state’s most adept Republican fund-raisers and became close friends with the first President George Bush and his wife, Barbara, during Mr. Bush’s two Senate races and his bids for the White House.

The elder Mr. Bush as well as John Tower, the first Republican United States senator from Texas since the Reconstruction era, were two of Mr. Straus’s early political mentors.

“I had the great privilege as a very young man to spend time with both of them, so when I think about some of the more ambitious, kind of out-there Republicans today, I always do go back to those, if you want to call them, old-school pioneers and founders,” Mr. Straus said.

The icy relationship between Mr. Straus and Mr. Patrick led to the cancellation of some of the weekly breakfasts between top Texas officials. During the entire 140-day regular session, Mr. Straus and Mr. Patrick never met one on one, an extraordinary occurrence for the two Republicans who preside over the State House and State Senate. Mr. Patrick has taken a series of swipes at Mr. Straus as their battle over the bathroom bill has heated up, calling his school-finance plan “a Ponzi scheme.”

Mr. Straus has taken a more subtle tack but has grown bolder in publicly criticizing Mr. Patrick, including on the bathroom bill.

“Putting aside some of his invectives, I’m always happy to meet with the lieutenant governor, and with the governor for that matter,” Mr. Straus said, adding, “I also believe that too many of our state leaders spend too much of their focus and attention on divisive social issues, so if that’s what the meetings are going to be about, you can count me out.”