I'm an atheist so (happily) no longer have to have these sorts of arguments with most people. However, I'm going to give my most Christian-friendly response. The sort of things I used to say before I questioned my faith.
I will say, if you believe in a literal interpretation of the Bible then a) good luck working that out, and b) yes, you are going against your religion by supporting gay rights.
However, most reasonable Christians acknowledge that a lot of the Bible is allegorical and some of it (like the creation story) should not be taken literally. They will also often use the (fairly valid) argument that Jesus fulfilled the laws of the Old Testament and effectively rewrote them with 'Love God and love others' (obviously paraphrasing), which is why they don't follow kosher laws and things like that. This would make the Sodom and Gomorrah argument, as well as the Leviticus argument, irrelevant.
Being even more broad with their understanding, clever Christians (in my opinion) will also acknowledge that the Bible was not written by God. It was written by fallible humans who, even if they didn't intend to, wrote their own meanings into the text. For example, my mother is a Catholic yet believes that Paul was a sexist and thus doesn't like some of his teachings and does not treat them as divine.
Think about the time these people lived in, and the fact that their main aim was conversion. Nowadays, politicians are probably the most ambitious to convert people; people ask them difficult questions, and they try their hardest to make their policies reflect what most people already believe or want. If you were Paul, trying to convert Romans (mainly Jewish Romans who were strongly against homosexuality) to Christianity, you would almost definitely claim that they could follow Christ and still keep the morals they were brought up with. To be honest, I'm surprised there aren't more anti-gay passages in the Bible, and the fact that there aren't makes me think Jesus probably said sod all on the matter.
Even now I'm an atheist, I think the sort of Christianity God would be most happy with is liberal Christianity. The ten Commandments are pretty sound, so is 'Love thy neighbour', so why not just follow the general jist of peace, faith, love and truth? It seems so pointless to argue over the nitty gritty of anything when it causes so much conflict.
EDIT: To relate this back to the original point, I don't believe people choose to be gay but, even if they do, the above reasons mean you can still support them and be a good Christian regardless.