The reason why the LGBT movement has proven such a challenge to organized religion is because the conflict has wrongly been interpreted as religion vs secularism when it is actually religion vs religion.
Gender identity is a faith-based belief that contradicts provable reality and as such it has no business being anywhere near secularism. Sexual identity, on the other hand, does not currently contradict provable reality because it is only an interpretation of it, so believers in sexual identity have the right to argue for its usage within secular society*, but organized religion has an equal right to be part of that conversation.
This is not creationism vs evolution in early 20th-century America. This is the Huguenots vs the Catholics in 17th-century France. The Huguenots stood little chance against the state-backed religion in their day, and we stand as little chance today unless we correct our understanding of what is actually going on. Organized religion can coexist with science, no problem; but organized religion cannot coexist with a religious movement that misrepresents itself as science in order to gain the official support of the state and effectively become the state-backed religion. We’ve seen this conflict before. We know how it will end.
* [Update 06/16/2021] I went too far in this prior statement. Yes, sexual identity is less egregious a violation of secularism than gender identity because at least it doesn’t directly contradict provable reality like gender identity does, but sexual identity is still an attempt to create an invisible construct that people are expected to believe in based on nothing more than faith in someone else’s words. Yes, you have the right to try to persuade our culture to treat the presence of same-sex attraction as a meaningful part of someone’s being, just like I, in contrast, argue that its presence should be treated as meaningless (because it is). However, you do not have the right to demand everyone believe a truth-claim about yourself that you cannot prove, and sexual identity therefore can never be legitimately used as a way to categorize people within secularism unless empirical evidence is discovered to prove its existence and thereafter that empirical evidence is used to identify everyone’s sexual identity instead of self-identity. But that will never happen because sexual identity, like vampires and werewolves, doesn’t actually exist in provable reality. It is nothing more than a faith-based interpretation of that reality, and everyone, therefore, is justified to reject it in favor of a better interpretation.