I'd be surprised if there were sound arguments for the immortality of homosexuality, but I agree with your suggestion that whether or not LGBT persons are 'born that way' or not cannot provide a sound basis for the immortality of homosexuality -- nor can it provide a sound basis for its moral permissibility of homosexuality either!
Your remarks about pedophilia suggest why such arguments are unsound: That a person is born in some way does not imply that actions they perform because they were born that way are not wrong. If (as seems likely) pedophilia is harmful to children, that it is wrong even if pedophiles can't refrain from having sexual desires directed at children. 'He/she was born with property P; he/she does X because he/she has property P; therefore, X is not morally wrong' is not a valid inference.
But perhaps this misunderstands the force of the 'born that way' claim. Perhaps the force resides not in the idea that being 'born that way' makes a person's actions morally permissible but that being born some way excuses a person's actions. So the thought would be that if homosexuality is inborn, then engaging in homosexual acts is morally excusable. But notice that (1) the reasoning above indicates why this doesn't seem obviously right -- being 'born that way' doesn't always make the actions one performs morally permissible, and (2) this strategy seems to assume that homosexuality is wrong but should be excused because it is inborn. Notice that for this strategy to work then, we would need an independent argument for the immorality of homosexuality. After all, you can only morally excuse what needs excusing, namely, actions that are morally objectionable.
In general, popular moral discourse greatly overinflates the importance of whether a trait is chosen or inborn to whether or not actions motivated by that trait are immoral or not. In the case of homosexuality, its moral standing must turn on familiar moral considerations (harm, rights, etc.) -- not on whether LGBT persons choose that way of life or are bequeathed it by nature or nurture.
(John Corvino, the best known philosophical defender of gay rights, discusses the 'born that way' problem here: http://www.huffingtonpost.com/john-corvino/born-this-way_b_3111186.html)