"...invited by our Archbishop...."

I'm surprised your Archbishop invited him! Cardinal Cipriani is a member of Opus Dei and by no means an admirer of Pope Francis . (This must have been before Francis' election.) Cipriani is so unpopular in Peru, especially among his own bishops. I know currently he will no longer be Grand Chancellor of the Pontifical Catholic University in Lima owing to any number of negative initiatives and a longstanding dispute between him and the university. Under a new statue, the Grand Chancellor will a bishop appointed by the Peruvian Bishops' Conference (or by the Pope). 
But of all people to invite to an American cathedral!! Good Lord! Horrible! Anyway, probably it's better you are with a Protestant group, church, especially among Episcopalians, who generously welcome LGBT people to their services, sacraments. I presume, too, there is no Holy Redeemer Catholic Church in LA (as there is one in San Francisco), which primarily ministers to LGBT Catholics.