"If he picks, I will have trust issues with the girl. If I pick, I won't, because I picked her."
I think you're making a big leap, to assume you won't have trust issues if you identify the girl that you want the two of you to date. In situations where a new person gets involved with a couple, they almost always form a stronger bond with one member of the couple than the other. It's just human nature, no two relationships happen in the exact same way, at the same time, at the same rate. Since your bf is the one who is actually into women, it seems likely that a woman would form a stronger romantic bond with him than with you. Would you really feel no jealousy in that scenario... or would you just feel betrayed because she was supposed to be "yours"?
"I feel like, due to my being straight, I would just end up one end of a V, and I would absolutely not be okay with that."
How could there be any other result? Isn't the very definition of being straight that you fall in love and in lust with, and want to have relationships with, people of the opposite gender? In a triad all three people share relationships with each other, how could that *possibly* happen in this scenario? Not trying to be mean, but I really don't get it.
I would strongly encourage you to just say "no" and to move on if he can't accept being exclusive. Alternately, maybe do a bunch of reading and soul searching and try to get more comfortable with the idea of a vee and consider finding another guy to date in addition to your bf... that's a much better way to avoid feeling left out than trying to date a girl when you don't even like girls and trying to stake some claim on her that almost certainly won't work and will just leave all involved hurt.
This is a tough situation. :/ Good luck!