I don't agree with my wife choosing a hospital to birth our first child
Could you explain this further? I'm curious why you don't agree with your wife's hospital selection.
Because we can't afford the one she chose and I'm worried we'll be broke
While it is understandable that you have concerns regarding the medical expenses of delivering your baby, you should be open to compromise with your wife since it is an important medical decision that affects you both. Since your wife will be the one delivering, she may have concerns about the facilities that you might have discounted. It is also worth considering that the initially more expensive hospital may end up being cheaper if it prevents an expensive delivery complication.
I'm going to go behind her back and call around for prices.
This is your wife and she is the one having the child, you really should let it go and let her pick the hospital. She's doing all of the work here.