its a flippen ONE year's birthday! it is NOT a party, it is a gathering of close friends and family!
I am pretty sure that once you issue invites, have food and a cake and specifically mention the giving or not giving of gifts, it has become a party whether you call it that or not.

I dont think that there is a polite way to tell your friend that her family is not invited, I also dont think that it is rude or out of line for her to assume that her kids are invited too. I am pretty sure she would not make the same assumption if you were throwing a hens night.

Good luck