I’m personally familiar with some of the principals behind Rainbow Railroad. They intentionally keep a very low profile in order to facilitate their rescues. I doubt they’d ever feel comfortable enough putting out enough public information to be rated as a charity. They actively avoid publicity and concentrate on person-to-person rescues.
Something else to be aware of with respect to persecution — the UN maintains refugee camps in some countries for displaced and persecuted lgbtq people. One camp in Kenya was in the news this week because of an outbreak of illness. Some of the conditions in the lgbtq refugee camps are really dire.
Contributing to charities that support those camps is a very good way of helping people who are fleeing violence for their sexual orientation or gender identity.