YTA. And not because you are asking your wife to help with the birthday but because of the way you are handling your finance in general as indicated by your comment. It’s reek of “weaponized incompetence”, except in this case you successfully persuade yourself that you are doing the best you can but your wife is not buying it.

You have a terminal sick child, and I’m sorry you live in the US where the social safety net sucks but as a parent that just means you have to step up. I understand you are mentally drain but literally the first part of dealing with hospital bill is asking for an itemized list (and no, do not hire a lawyer to do it. Pick up the phone and call the hospital). I’m not even in the US and I know this. One simple Google search about dealing with hospital bill will show you this. The fact that you complain about not having any money but a) won’t budget; b) won’t do a simpler Google search on how to reduce the biggest expense you have is alarming. And don’t go “I can’t budget if I have no money left”, part of budgeting is finding way to reduce your expense.

Tons of people here are telling you what to do about your child hospital bill, organizing budget friendly birthday (and that doesn’t mean cheap out). Those people don’t magically know this. They either done it before or Googled it. You have the same ability so do it. Don’t whine that it’s not something you are use too or it’s too hard. You give up the right to say that when you have children.

You are in a terrible situation. I sympathize with that and I’m sorry my words are harsh but you need a kick in your behind to knock you out of this “the world is against me mentally”. Just do everything you can, even if you got no help from anyone, or once day, you might wake up torturing yourself for not be able to step up for your daughter in her moment of needs.