NC because potentially outing. I will try and keep this fairly simple. There are some extra details which I won't go into for now but there is no backstory of any rows with SIL, who we have always got on OK with until now. BIL is DH's brother and SIL is DH's brother's wife.
DH's brother and his family live in the same town as us and until this year we saw each other regularly. DH and BIL have always been very close and our DC are close in age.
Our second DC was born in January and since then my SIL has not seen or spoken to us. We have seen my BIL and their DC but less frequently than usual. For a while we were busy with newborn life and had other things going on but after a while it started to feel really weird that SIL had still not met our DC or even congratulated us.
When DC was 6 months old DH confronted BIL and asked him whether there was a problem. It turns out the problem is that the name we chose for our DC2 is one of their DC2's middle names, which I knew beforehand, and so we should have understood that the name was off limits and we shouldn't use it. I did know this beforehand, but I didn't think it mattered, given that it is a very popular top 10 name, and that they gave their DC1 the same first name as another family member. Apparently we were wrong and it matters very much to SIL, who hasn't spoken to us since.
Their confrontation was about two months ago and we hadn't seen BIL since, but he continued to be chatty enough on family WhatsApp groups and like pictures of our DC etc. We are certain that all of this is coming from SIL and not BIL. Yesterday we were all invited to Sunday lunch at PIL's house and everyone was there except SIL. BIL's DC1, who is 3, came up to us and loudly demanded to know why we had stolen his little sister's name. This was glossed over in the moment, but last night DH sent a very angry message to BIL saying that it was not on for them to be saying these things about us in front of their DC, that SIL's behaviour is very childish and that no one owns names, especially not middle names.
DH is very upset and so are PIL. There has been zero drama in this family for as long as I have known them.
Who is in the wrong here? And how do we go about fixing it?