From any standpoint, the theater performance, being scheduled first, takes precedence. The brain-farter should have simply told the birthday party "I'm sorry, there was an error, and the space you booked is not available at that time. Here are several options for rescheduling (options) and if none of those are acceptable, I'll happily refund your reservation."
In my opinion, she absolutely erred by even CONTACTING you.
I agree that the Theatre Group should absolutely take priority over the birthday party.
However, I disagree that it was rude for the Reservations Lady to have contacted the Theatre Group. I'm guessing that the Birthday Group were being stubborn donkeys, and refusing to move their party. So she called the Theatre Group on the off-chance that they might be able to move their show. So long as she didn't threaten or bully the Theatre Group into moving, I think she was fine.
Anyway OP, I'm glad it worked out in the end.
I disagree, I can't think of any long standing theatre group that would be able to reschedule a show on such short notice. Quite frankly, they'd have to be a pretty horrendous group to not have tickets already sold in advance for the show! And not necessarily have contact details for all of the patrons either if some tickets were sold over the counter, so even if they had the resources to it would be a big ask to contact all of the patrons and reschedule, plus damaging to their reputation. The suggestion of moving a theatrical performance that is completely scheduled and ready to go is just absurd IMHO. That would only happen in the case of a real emergency such as a the actors all coming down with the plague or the theatre burning down on opening night.
It doesn't sound as though the lady actually thought they might move the show, just that she thought they wouldn't mind "sharing" the venue. Which I guess she figures is "ok to ask" but really it was her mistake and she was making it their problem. Once they'd booked the venue, it shouldn't be their problem who else wants it - they have a commitment. Even asking puts unfair pressure on them. So I think it was a little inconsiderate of her not to just fix the problem.