I think generally Debbie Downer thinks that she's found what she's looking for in a relationship with a man when she hangs on to a man who's obviously never going to make a pass at her, and invests all HER quality time in a gay man, who may be an excellent friend but will never make a proper life companion. It makes her feel safe, I'm sure, that she is loved while not having to deal with the crap that real sex with men brings. Of course this is rather immature. Debbie Downer probably needs to grow up.
That being said, her position in the trio has now been compromised by the fact that your boyfriend seems to have found what he was looking for. She thinks she's found it but she's been barking up the wrong tree. It would make sense for her to go out and get a life of her own, except it is difficult to let go of a good thing, and a good friendship, especially for an insecure girl. She must be a bit insecure, from what you've already said.
How would you consider being her bbf for a while and trying to make her more attractive to OTHER men, the ones that she'd never consider trying to start a relationship with? It's not healthy for her to stay only with gay guys, if she wants a romance of her own. She needs to go out there and start dating, don't you agree? If you were to consider this, you might get your boyfriend onto the wagon with you and try to give her the makeover that she needs. Is she wearing the right clothes, is she giving out the right signals? Why is it she thinks she will never be loved? Is she ugly? If so, what are her strong, good, points that can be brought to the fore?
As for the boyfriend, I think it would be ok sometimes to try to limit the occasions when the three of you are together. You both need that quality time for yourselves. She doesn't sleep with you in bed when you have sex, does she? Well, that's how it should be for other times in your life as a couple.
Talk to your boyfriend, tell him what you're seeing. Admit that he must have some private time with her, but that you'd rather not be tagging along at those times. Give him the freedom to spend time with her. It's healthy that way. But also insist on having the same privilege, of not having her along with you. See above.