DENVER — Donald Trump’s momentum with gay Republicans grew Tuesday with the endorsement of the Colorado Log Cabin Republicans, the latest affiliate to defy the national organization’s refusal to back a presidential candidate.

The Colorado chapter joins the state group in Georgia and local chapters in Los Angeles, Miami, Cleveland and Orange County in backing Mr. Trump, despite the Log Cabin Republicans PAC’s decision last month to withhold its endorsement.

After some behind-the-scenes grumbling, however, the national branch gave its clubs the one-time option to make a chapter-based endorsement in the 2016 presidential election.

Colorado Log Cabin Republicans President George Gramer pointed to Mr. Trump’s outreach to LGBT voters, such as the nominee’s inclusion of Silicon Valley mogul Peter Thiel as a speaker at this year’s Republican National Convention.

During a campaign stop Saturday at the University of Northern Colorado, Mr. Trump posed for photos holding a rainbow flag with the inscription “LGBTs for Trump.”

“As a retired Army colonel, national security issues are especially important to me and I’m proud to give a shout out of support to Donald Trump on behalf of Colorado Log Cabin Republicans,” Mr. Gramer said in a statement.

“Hillary Clinton’s policies endanger the LGBT community and she has consistently put our nation’s security at risk for her personal gain,” he said. “We enthusiastically endorse Donald Trump for president because of his commitment to America first, peace through strength, and free enterprise.”

Jeff Davis, vice chairman of the Log Cabin Republicans of Texas, said his state affiliate has not voted to endorse Mr. Trump, although “the current status is up in the air at the moment.”

The Log Cabin Republicans PAC described Mr. Trump in its Oct. 22 non-endorsement as “perhaps the most pro-LGBT presidential nominee in the history of the Republican Party,” but expressed concern about Trump advisers with “a record of opposing LGBT equality” as well as his support for the First Amendment Defense Act.

“Log Cabin Republicans have long emphasized that we are not a single-issue organization, nor are our members single-issue voters. Even if we were, rhetoric alone regarding LGBT issues does not equate to doctrine,” said LCR PAC in its statement.

Kenneth Wilkison of the Colorado chapter cited national security concerns, saying, “we don’t want to have Islamic terrorists endangering our lives and safety, we don’t want to die.”

“Hillary Clinton’s policies are lethal and must be stopped,” Mr. Wilkison said in a statement.

Mrs. Clinton’s once-double-digit lead in Colorado has slipped in recent weeks. A CBS/YouGov poll released last week showed the Democratic presidential candidate ahead by 42 to 39 percent in the critical swing state.

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