The most prominent LGBTQ+ republican organization is call the Log Cabin Republicans and annually present the "Spirit of Lincoln Award". They were founded in the '70s so the idea that Lincoln was LGBTQ+ himself has been around for quite some time.
As a side note, recent recipients of the "Spirit of Lincoln Award" include Melania Trump and Kellyanne Conway who are obviously prominent LGBTQ+ advocates...
Lots of very full leopards around the Log Cabin Republicans I think...