Did you tell your sil at the time that you had already got plans and that MIL had already promised to babysit for you on that evening or were you so gobsmacked when she told you that you just stood there going errrrrr?
If you just said errrr (or words to that effect
then you are probably stuffed as she will have taken that to be a yes I agree to your plans.
if you did say something - would love to know if she actively said she was cancelling your plans
have you actually spoken to mil about it to see what she thinks - has she even said sorry about having to cancel on you but I really want to be there so howabout I babysit a different night (although I know, it's not the same if you had made plans well in advance for it to be the night)
Of course, what you should do is to tell SIL that as your plans came before hers then the fair thing to do is to have the kiddie party on the saturday afternoon, but have the adult party on the sunday, and that you have sorted it all out for her 
not that I would actually expect that to happen, but it would give you a chance to ask her how she likes it when somebody else changes plans that you have made! And maybe she could see that she is being a little unreasonable to make changes without asking all of the people involved.
Alternatively you could just say fab, dc will be at the party, you won't all mind looking after them will you and dh and I can nip out for our pre-planned birthday meal.
did you actually have a restaurant booked yet that you could dangle in front of them to make them feel even guiltier?