I think there might be a philosophical difference between you and Bee. It feels like she sees her and Jay very much as two separate individuals. They share what they have decided to share, and some of those are big things like childraising and home. But she still seems to want quite separate social circles.
I feel that you might see them as more of a unit. I don't mean that you don't think they are individuals, too, but maybe there's a scale where you think of them little more towards couple while they see themselves more as individuals.
Say, Jay had a very close best friend, who met you and liked you and you get along fine, but they still preferred to hang out mostly by the two of them. Would you feel it essential you and him become closer so that it would be comfortable spending loads of time the three of you? Or would you be able to accept that even though he is a really impotant and close person in Jay's life, you and him don't need to be particularly close?
Even if two people are married and seem very couple-y, they still may have an outlook where own close people don't become the other's close people. It can go for romantic partners as well as friends.
Could you see you and Jay building a true partnership which is mostly separate from his life with Bee? If you and Bee don't become closer, does that make you feel threatened in you relationship with Jay? How about you work on that and try to trust that Bee is satisfied with the situation with Jay having you in his life, even if she doesn't have time or energy to build a close friendship with you? From what I hear, she sounds like an introvert, and some of us need loads of time and space for ourselves.
Oh, and one more thing. I don't think you are doing anybody any favours by acting less close to the baby when Bee is around than you actually are! Please, stop doing that. I feel that it comes from a place of fear (fear that she may be uncomfortable with that and then choose to deal with that discomfort by getting rid of you?). You need to trust that she will work on her possible feelings of discomfort. If they are set on poly, she should, and it is very likely she will. However, if her coping mechanism is getting rid of the new person, you won't be able to ensure 24/7 comfort and protect yourself from that. Also, you can hopefully trust him. Have you talked to him, if Bee asked, would he happily dump you rather than expect her to get over herself?