I've NC for this as your story sounds very similar to mine, and I don't want my other profile to be found as a result of what I'm willing to share.
15 yrs ago, XH and I had our first DC. The birth was traumatic (EMCS) and our baby v poorly (he's a fine strapping young lad now - rest assured) to the point where we were asked to name him. The only name we managed to agree upon was (and this isn't exactly it but you'll get the gist): William. So that's what we named him. He pulled through, thankfully, albeit there was a lot of time spent in hospital and there still is. I'm sitting in hospital with him right now hence why I felt compelled to write. I spent his first two weeks living in special care, came out for 3 days and we were re-admitted almost immediately. It was all very complex and complicated, and I wasn't prepared for any of it. XH was useless.
3 months later things had improved but slowly, and we went to a family party. Saw XSIL there - hadn't really heard from her despite the issues we had faced. At the party, she exclaimed we had "stolen" her name, and didn't speak to me for - well, to date. Her son who was 8 when ours was born (they live about 200 miles away) is called Billy. I was on the scene after he was born. He was christened William. Did I know this? Yes. Did I seriously think it would have caused such a schism? No. Would we have used William if we had telephoned her to ask to use the name, and she had said no? Under the circumstances of his birth and subsequent issues, it was the last thing on my mind at that moment.
XMIL refused to get involved apart from to say "you need your heads banging together." XSIL had a meeting with XH when DC was about a year old, and subsequently it came out that he "became so aggressive" she got very scared (I didn't see this side of him until the latter stages of our marriage in 2016).
All I can add to the conversation on this thread, is that PILs will want to maintain a relationship with all their children and indeed their grandchildren, because that's what matters to them - they don't want to have to take sides, or become heavily involved in such disputes as it will risk their own interaction with their children's children. If your PILs decided to support you, they risk having their weekly childcare provision removed, and that's not ok for them to have to do.
We didn't push it, we carried on going to the family events but we showed our faces and then left promptly. It was horrific, and hugely awkward at times, but we were civil and tried to make it as easy as possible for my XMIL. We were all invited on a week long family holiday together by XMIL when DC was 2, and it was so awkward, we left the following day.
I don't know what the relationship is like today, between XSIL and XH; but my DC tell me they rarely see their Aunt / Cousins on that side. William doesn't know this was ever an issue, which is the main thing: I've protected him from that at least.
I know it's hard and I understand why you feel so bereft - but clearly, some people have very strong views about ownership of a name. The irony is practically every male child in our family has the same name as my dad, his dad, and his dad before him - and the wonderful thing is, nobody bats an eyelid. It's a family name and all the children grandchildren and great grandchildren each have their own nicknames, I'm even married now to someone of the same name. You can imagine the number of heads which turn when the name is called out at a family gathering 🤣. It's the tradition of our family: it is not the tradition of XHs, and I have had to come to terms with that.
Good luck with whatever you choose to do - try to take a few steps back though, this isn't really your responsibility to fix.... you can't change the behaviour of others, you can only change your reaction and your response to their behaviour. 