Stay home and have a small party - stick to your budget!
Your daughter will not remember her first birthday anyway,
but in the moment you can make sure she has a fun time without overdoing it.
Plus a few pictures to look back on later are always nice.
If you're going to have a lot of guests and/or other children there, you can still have a nice time. How about suggesting a potluck and everyone brings something to pass or share? Hot Dish - Salad - Dessert - Paper Plates - Napkins, Plastic Utensils etc. Nothing wrong with having people chip in for a celebration!