By Ginger Gibson and Emily Stephenson

CLEVELAND (Reuters) - U.S. House Speaker Paul Ryan offered his strongest support yet for Donald Trump's White House bid on Tuesday at the Republican National Convention, saying the New York businessman he has repeatedly criticized provides America "a chance at a better way."

Behind the scenes in Cleveland, the highest-ranking elected Republican has been shopping a competing vision for the party, backed by some wealthy donors and Republican establishment figures who see him as the savior of a party thrown into turmoil by Trump's candidacy.

The support reflects deep divisions within the party over Trump's harder line on stemming illegal immigration and his threats to scrap trade deals that he says have hurt American workers but establishment Republicans view as central to their free trade policies.

The split has created an identity crisis for the party, embodied in the figures of Trump and Ryan.

"Everybody knows the tough spot Paul Ryan is in," said Annie Dickerson, a New York delegate to the convention and close adviser to Republican financier Paul Singer, referring to Ryan's dilemma in backing a candidate as the party's standard bearer in the Nov. 8 presidential election who does not represent many of its traditional values.

"He’s the antidote to all the frustration that’s in Cleveland this week," she said, noting that despite a public push for unity, there are those still unhappy with Trump's nomination.

Ryan, 46, is chairman of the convention at which Trump was officially nominated on Tuesday. But he has kept a low profile, avoiding the convention's opening day on Monday, and instead holding meetings with state delegations and addressing private gatherings of Republican supporters, according to aides and attendees.

At the meetings, according to attendees, Ryan has been pushing his "Better GOP" platform, a six-part agenda that shares strong similarities to the form of conservatism President George W. Bush advocated - reduced regulation, lower taxes, and welfare reform. It contrasts with the speeches typically delivered by Trump, who prefers to lambaste his opponents and skip detailed policy discussions.

Doug Heye, a former House Republican leadership aide close to Ryan, said Ryan's platform "gives them something that they can talk to constituents about that isn’t divisive, that isn’t name calling."

POLITICAL SUICIDE

Heye said the approach could help Ryan position himself as the face of the party if Trump loses the election.

Ryan has publicly clashed with Trump over the latter's promise to deport millions of illegal immigrants and end trade deals. He criticized Trump's call to temporarily ban Muslims from entering the country and called his remarks about the heritage of a Mexican-American judge racist.

Ryan supports free trade and comprehensive immigration reform, and as chairman of the House of Representatives Budget Committee, he authored a document that called for a drastically reduced federal government.

The risk for Ryan is that a swath of the party is moving away from traditional Republican orthodoxy, and by not embracing Trump he could find himself isolated. Some of Trump's supporters see him as emblematic of the Washington dysfunction that has angered many Republicans.

Mark Bolin, a Canton, Ohio, business owner who supports Trump, called Ryan’s tepid endorsement of Trump “political suicide.” He said Ryan and other Republicans leaders can’t admit their own failures.

"Instead of them thinking there’s something wrong with their party, they think it must be us, the American people aren’t smart enough," Bolin said.

Ryan has long been seen as a rising star and some wealthy donors pushed Ryan, who was the vice presidential nominee in 2012, to run for president this year. He declined but hasn't ruled out running in the future.

Advisers to Singer and the billionaire industrialist Koch brothers say the donors are banking on Ryan to carry the flag for traditional Republican values.

"I like Ryan and his day to lead the party will come - very soon, if Trump loses," said energy magnate Dan Eberhart, another major donor.

Linda Devore, 66, a retired attorney from Fayetteville, North Carolina, who met Ryan for the first time in 2012, rushed over to give him a hug when he arrived on Tuesday morning to speak to her delegation at a hotel in the Cleveland.

"He’s just one of the smartest Republicans. That’s what I told him,” Devore said. “He’s a quick wit. He said, 'Pretty low bar, huh?'”

(Reporting by Ginger Gibson and Emily Stephenson.; Additional reporting be Amy Tennery, Emily Flitter, and Michelle Conlin; Editing by Richard Valdmanis and Ross Colvin)