If you were inviting friends and asking them to pay for their own pizza, I can totally see how that would be classified as rude. But, that's not the case here. Your family made plans, if other people choose to join in, that's on them. Tell them, "DD asked to just have pizza at ______ for her birthday. If you'd like to meet us there, you're welcome to, but you'll have to cover your own meal." You're doing what your DD wants for her birthday. If they can't accept that, that's their issue.
Hi! I'm Shannon
Wife to J
Mama to a couple of crazy boys - L (12/07)
and E (2/11)
and Meatball coming 4/14