My impression is that the Log Cabin Republicans aren't very good on trans rights. Although they haven't dropped the T from the LGBT acronym, their inclusion and support of trans people seems to be qualified and limited. Last summer, the Log Cabin president said that the onslaught of anti-LGBTQ legislation (currently over 500 bills in 2023) does not amount to an "emergency" situation, and he dismissed the set of issues in those bills as "trans surgeries for minors, biological men competing in women’s sports, and sex and gender identity lessons in kindergarten," a characterization that seems inaccurate as well as unhelpful.
Also, I don't believe Log Cabin has any trans people in leadership; of the nine Log Cabin leaders who have bios online, none include the word "trans" or "gender." If the organization isn't being co-led by trans people, that helps explain why its president doesn't seem to really know or care what trans people's issues are.