FS - that sounds very very hard with your 13 year old. And worrying - for it to continue for a month to the point where she is losing weight is also very difficult. I don't see what else you can do other than what you are doing - especially in these circumstances - and I think reaching out for help from your MIL and your H - which you have done - is also the right thing to do.
I am sure you've thought of it, but I wonder if your 13 year old is reacting to everything that's happening in the world right now - feeling afraid and out of control and taking it out on the closest, safest person? I guess you have to be really careful - as you are being - that conflict over parenting methods doesn't become a proxy for other conflicts between you and your H. (I am saying this to myself as much as you - my son is a little older and while he has meltdowns now and again that aren't age appropriate at all, his behaviour day to day isn't as worrying as your daughter's is). I know my son has in the past become a kind of symbol of respect and control between my H and I, and I am worried in these circumstances that is creeping in again. I don't trust my H to do what is best for him, I think he's motivated more by wanting to be right, wanting to 'win', wanting to get his version of respect. Do you think there's any of that doing on between your H and your daughter, or you and your daughter?
This is very very hard. And our options are so limited right now. What helps when my son is in a sulk is getting the dog involved. Is there something your daughter really loves that you can tempt her towards you with? I'm sure you've already thought of that.
Hang in there!
Edited to add: I guess what strikes me is that you say your H is taking 'her side' on a lot of things. I think my H feels like that too, in conflict between him and our son. I wonder if it might help both of us to take the idea of 'sides' completely off the table, and try as parents to come to some agreement as to what is best for your daughter / my son. As I said, I find that nearly impossible with H when he's in certain moods, and he is of the type who is the world expert on parenting having never read a parenting book, spoken to anyone else about parenting, or considered any other techniques - so it's a very very difficult conversation to have with someone immovable. Is there any agreement or common ground you can seek with your H? Does he have any suggestions for you that you haven't already tried?