If it's any consolation to you and your sensitivities about the LGBT community, many lesbian and gay people support same sex rights as it underpins the very notion of same sex attracted people. I am a lesbian and I am relieved that men who identify themselves as women will not be able to access female only spaces. Being a woman is supremely important to me, my politics, my spirituality, my sexuality, and the material reality of biology matters. Indeed, there are many trans identified people who believe in same sex right and spaces - Buck Angel being a notable and vocal female who lives as a man (her words) if you are interested in looking at the reality of people in the community.