From the Wikipedia article on Log Cabin Republicans
The Log Cabin stresses its loyalty to the Republican Party: "We are loyal Republicans," its website says. "We believe in low taxes, limited government, strong defense, free markets, personal responsibility, and individual liberty. Log Cabin represents an important part of the American family—taxpaying, hard working people who proudly believe in this nation's greatness." They take no position on abortion.
But Log Cabin dissents from socially conservative Republican views on matters relating to gay and lesbian rights. "We also believe all Americans have the right to liberty, freedom, and equality," it says. "Log Cabin stands up against those who preach hatred and intolerance. We stand up for the idea that all Americans deserve to be treated equally—regardless of their sexual orientation."Log Cabin's criticism of President Bush
The 1992 Log Cabin Republican convention was held in The Woodlands, Texas, a Houston exurb. It was the first time LCR had such a large presence of the major national press at a convention. The big issue was whether or not LCR would endorse the re-election of President George H. W. Bush. The group voted to deny that endorsement, based on the 1992 Republican National Convention that preceded it in Houston.
Although Log Cabin supported the election of George W. Bush in the 2000 presidential election, in September of 2004 LCR's board voted to withhold its endorsement of President George Bush's re-election, Bush's support of Federal Marriage Amendment being one of the significant reasons,..."
Well, maybe if the Republican Party splits, they can stay with the moderates and have some influence.
I think I would accept the invitation so that I could ask why they try to influence the Republican Party on Gay issues, instead of influencing the Democratic Party on their other conservative issues. If they helped get the Dems elected, at least they could achieve their Gay rights issues.