I've just gone back and read the entire thread. It just gets weirder the more I think about it.
When I was expecting LO I did consider Isobel or Isobelle as a name but then realised it's a form of Elisabeth, the name we gave to our daughter who died, and that was too close for comfort for me. LO was a boy anyway but my point is there are similar names that have the same meaning, which your SiL could have chosen instead. She could have had Elizabeth Ava if she liked the style of Isabella Eve (for example). Of if she had to have a Bella why not an Annabelle or an Arabella so at least they had something that was obviously different, even if the nicknames were the same?
I know a lot of people have said that Bella doesn't make sense with Isobelle but there are lots of nicknames that don't make sense with the original. I deliberately chose my daughters name with an 's' rather than a 'z' to avoid Liz or Lizzy but had she lived I'm sure someone would have called her one of them. But either way, 's' or 'z' there are lots of odd nicknames for Elisabeth that don't fit, Bess or Bessie or Betty for a start.
And Margaret is/was often shortened to Peggy and I have no idea why.
Not sticking up for your SiL BTW, just saying that sometimes the nicknames don't fit and in a normal situation it wouldn't matter. Your situation is far from normal.
I can understand why you don't want to be the one to change either. Your Bella was here and named first, you've been using the name for months, even before she was born. No doubt you have congratulations cards with Bella written on them or gifts that have her name on. Everyone knows her as Bella. Your Bella knows her name by now. SiL's Bella is less than a week old and so they won't have all that yet, so it's not unreasonable to think they could change it a little bit or use Izzy as a nickname as her baby won't know it yet.
And at the moment its obvious that your Bella is the older baby but when they are a little older, both three or five or ten etc, it will be hard for people to know which Bella was first, so you can't even take comfort in being original.
It's not only a name, it's three names, first, middle and surname, and they are bloody identical. It's weird. I agree with whoever posted to say she had done it on purpose to get at you somehow. It's a passive-aggressive way of going about it, and a totally mental thing to do.