Another thought popped into my head here, which would apply to any religion that both believes that (1) gay sex and gay marriage is "sinful," and (2) one should evangelize to spread the word. Suppose that if I converted pursuant to being evangelized, mental health professionals having "examined" me, opined that the odds would then dramatically increase that I would become desperately unhappy and a neurotic mess (say even materially more likely to commit suicide). Would it be a sin under Catholic theology, or whatever theology of another such religion, to evangelize me under such circumstances? Is there any concept at all, that a given theology is not for everyone given their situation?
As someone who belongs to a different church that meets the criteria you outlined, our point of view is that
everyone should be evangelized. We all bring different struggles to the table, and we may be happier in the short-term for satiating them, but our souls will benefit in the long-term if we abide in the spiritual life of the Church and fight temptation. There are many sinful tendencies in my own heart which I have had to deny. It's been hard, but as I have worked through it I've become a happier and more centered person as a result.