Ok, that may actually CHANGE a little bit of the info and help you may recieve. (Of course, I may be the only one here who didn't pick up on it. LOL)
Anyway, a relationship between two people is always difficult. Add in a third or even a fourth, and the problems compound exponentially. Not just a little, but a LOT. Now, you have to remember, that if H is having an issue with your girl seeing another girl (you), his issue is likely this: "Man...I'm dating a lesbian, and I can get her to like men! Yeah! But if she's going to continue dating girls, then I can't get my groove on and switch her around."
Now, this is just a POSSIBILITY. We men are weird beasts. For some odd reason, we think that we can somehow make a bi-sexual woman or a lesbian, like men enough to ONLY like men. We think that we can re-wire them and "fix" them. Well, ok, SOME men think this. I wouldn't try to, but hey...Some guys think that.