First off, thank you so much to everyone for the incredibly thoughtful responses. You've given me a lot to think about, and I still am working through it.
I'm noticing concern for Bee's side in all this, which I mostly left out in my first post for the sake of brevity. I think some background would be useful.
Jay and Bee have known each other since they were 18 and have been married for almost 10 years now. They opened up their marriage about 6 years ago, the impetus being that it had turned into an essentially non-sexual relationship. Jay dated his first OSO for a couple of years, and they're still friends. It sounds like they did more things together as a group, although Jay considers her mostly his friend, and not really Bee's. Jay and Bee don't really socialise together outside of family. Bee has a long distance girlfriend and she recently started a new relationship, also long distance. At this point they've expressed being pretty committed to being poly. They're very committed to each other, and their marriage is stable (from my perspective, anyway).
I've previously expressed worries that Bee might feel resentful of me or that our relationship negatively impacts her somehow. Jay tells me that she feels positively about our relationship because she's relieved from having to think about Jay's sexual needs. In terms of balance, and especially in terms of childcare, they both feel that they're putting in (and getting) a fair share because Bee goes away on trips every month or so, sometimes for four days, sometimes for a week or more, and Jay gets to spend time on hobbies and on me when she's in town, but takes care of everything on his own when she goes away. Also, Jay is responsible for almost all the house maintenance, housework, and dealing with contractors.
I'm told by Jay, and to a lesser extent by Bee, that she doesn't believe in getting involved in Jay's relationships. She feels very strongly that he shouldn't be asking her for permission, and that the metamours should not feel pressured to be friends. I agree entirely with those sentiments, but I also think that being able to communicate directly would make things easier, and NOT being able to do that gets increasingly awkward with my increasing involvement in Jay's life.
What I want is for us to be open to the possibility of being friends. I don't want to force anything, but I want to create situations where it will be possible, rather than the opposite. In terms of concrete steps, last week I suggested that Jay organise a group activity about once a month, which he agreed to. I feel nervous about Bee being actively averse or even just completely disinterested to growing familiarity between us. There's been a pattern of avoiding opportunities for group time - we haven't planned any group time explicitly, but there have been plenty of opportunities for family dinners, activities with mutual friends, etc. My contribution here has been to adjust my expectations for socialising and convince myself that Bee isn't avoiding me out of personal dislike, she's just not very social generally. Which is what I've been told by both Jay and Bee and I will not second-guess that.
I feel (and Jay and Bee, apparently) that Bee is getting her needs met in their relationship. Jay and I have spent a LOT of time talking about what Bee's needs are. Although Bee is pretty closed off to me I'm confident that she and Jay have good communication patterns. She was the one who initiated the scheduling talk, which was actually very brief, because she already knew what she wanted, and I think included me mostly based on principle/courtesy (which I do appreciate). We acknowledged that scheduling had been getting out of hand over the last couple months, and she suggested that Jay be home two nights a week and that "Sundays when she's not working are important because they're the only baby-free time they get". We all expressed agreement and then we made a shared google calendar and scheduled the regular time in.
I think it's important to build familiarity with Bee because family is such a huge part of Jay's life right now, and getting closer to Jay seems to mean getting closer to his family, or at least it's the direction that things seem to grow towards. I don't want to be in a relationship where I have to continually deny the instinct for closeness because it manifests in an inconvenient way. If Jay and I had the type of relationship where we just went on dates or out with friends together then it wouldn't matter, but it seems like we continually gravitate towards sharing "home" type intimacies. One way of intepreting the pattern of our relationship so far would be that we get closer, Bee tries to make room for us by avoiding us, but avoidance isn't sustainable, so there's some friction, we pull back, and then things eventually get easier.
For example, there was a period of time where I was sporadically co-sleeping with Jay and Baby (they have separate rooms and alternate nights). After a few times of that happening, Bee told Jay that she sometimes felt excluded when Jay and I were taking care of Baby (although she recognised that we weren't actually excluding her), and that co-sleeping specifically was triggering for her. We stopped doing that, and a month or two later she said that she was now OK with co-sleeping. This pattern results in feelings of unstability and insecurity for me though. I would like it to be a process of slowly and collectively expanding our comfort zones. At this point I am incredibly intimate with Jay relative to how distant Bee and I are, and the imbalance is uncomfortable.
I'm generally used to checking in with Bee's needs. When I make plans with Jay I'm careful to make time that doesn't conflict with Jay and Bee's schedule, especially Bee's work schedule and the available babysitting time. When I visit at their place I often entertain myself (or myself and the baby) and Jay runs errands for Bee or does household stuff. I'm also conscious of Jay's needs (to a lesser extent, because he's not as vocal), and a lot of the time we schedule is time that impacts him minimally (lunch break, company when Bee is away and he is stuck at home with Baby, etc).
As for Baby, we all recognise that consistency and attention for him is completely non-negotiable in this dynamic, and I recognise that I have to balance that with my own relationship with Bee. When Bee isn't around, my interaction with him is pretty close. I've grown used to changing his diapers, feeding him, dressing him, bathing him, knowing what his favourite toys and games are, how he likes to be soothed, etc. I've gotten to feeling really attached to him and thinking about it actually makes me kind of emotional. I'm holding it back because it's impractical but I don't want to have to hold it back forever. When Bee is around I hold back a LOT, and whenever possible I try to defer to her or ask her how she wants things to be done. Basically, with Jay, the dynamic has evolved into that of familiar nanny/aunt, but with Bee it's more like the visiting friend who doesn't really know how to interact with the baby.
I'm not sure where this is going and I want to wrap up this post. I hope the additional information has fleshed out the context of this relationship somewhat. Presently Jay and I are somewhat in limbo and unsure of how to proceed. I'm starting to feel like we should try to "start over" and hold back on any family-type intimacies unless Bee is genuinely comfortable with them, and not just comfortable tolerating them temporarily. But I think that for that to happen, Bee needs to be interested in getting comfortable, and that can only happen if Jay is able to express that it's important to him. He has said that it's important to him that I don't have to avoid Baby or his extended family or visiting his home, but he hasn't been able to express this in terms of what level of involvement he wants. I think, mainly, it's just important to him not to have to avoid those things. To me, I can't turn the attachment on and off. I too want to be involved in those things but I need consistency.
I've tried letting Jay know what would help me feel more valued in this relationship. I've given examples like making plans with me (I almost always initiate), physical tokens, verbal affirmation, examples of communication patterns that make me feel more or less understood. He's well-intentioned but also very absent-minded and has a lot on his plate, and it's been inconsistent. Yesterday I asked him to make daily contact, at least for the next couple of weeks. Things have been volatile for long enough that I don't feel like I can count on anything - sometimes I can't even feel sure that we are in a relationship. I think the daily contact will help establish some stability - I don't think I'll need it indefinitely, but it goes a long way for now. He agreed with very little fuss (which actually surprised me pleasantly), and I'm hoping that a couple weeks of this will help build my confidence in our relationship.
Thoughts, feedback, will all be appreciated. Reading your posts has forced me to think things through more clearly, and writing things out definitely helps too. I've been feeling a little more clear-headed and self-directed about things.