But presumably your future MIL knows your parents don't do Christmas, from the year she invited your brother: therefore, she has had the opportunity to invite him and hasn't. I think your fiance has worked this one out, hence his comment that there's not space or whatever for him: it might be FMIL has already made a comment to him.
It's not like your parents aren't here, or aren't celebrating Christmas: it's not their tradition/culture/whatever. So whilst I understand your brother had a lovely time the year he was invited, I can see she doesn't want to be obligated to invite him every year - otherwise, surely she'd already have included him. If you ask her directly, you're putting her in a bad position - who could feel reasonable about refusing to host a child for Christmas in that position? Again, I stress- this isn't a child who celebrates it every year, and would be missing out otherwise: this is a child who had one Christmas, once (and not even last year) where it is not part of his family's tradition.
We OP is married, then she can have Christmas at her home, and invite her brother every year, if she pleases - but I think she's being unreasonable hoping to inviegle an invite for a child who has had one Christmas, but whose family don't ever celebrate it, when his parents are at home as normal.
I think her boyfriend is livid because she's already accepted the invite, confirmed her attendance, and is now considering not going to spend the day with her brother and her family - because it's Christmas, so her brohter doesn't feel left out, even though her family don't and never have celebrated Christmas.
I get that it's hard being somewhere where the majority of people celebrate different traditions to you - and sometimes, it can be very lonely. But this is a pre-teen boy, with a family, in a home with them, where they do not celebrate - not quite the same as an adult living alone.
Sorry, OP, but it's not a dilemma: you accepted the invitation, you attend - and if you really need to see your brother on the Christmas neither he nor you traditionally celebrate, then find a time its not inconvenient for your hosts and pop back the ten minutes you've said it is.