I don't like free bar weddings truth be told. There's usually some idiot who gets too drunk for everybody's good and lots of people take the piss, ordering double and triples and expensive wines they wouldn't if they were paying.
I'm shocked at and would totally uninvite from my imaginary wedding the poster who says cut your cloth by inviting less people and do free booze!
it sounds like they're only inviting a few due to cost anyway, imagine having even less people at your wedding than you could have afforded but made the choice to fund fewer people getting pissed!? I'd rather have an invite and buy my own any day off the week and so would your guests I should think.
It does make sense to talk to the restaurant about how best to manage the drinks - some will take people's bank card, give then a number, and they can order drinks to their tab all night, others won't have that set up so will need to do individual bills to be paid cash or card with much to and fro of the waiting staff or customer - clarify that.
The tip will also be a "thing" to consider - can you afford that? Might smooth everything over if you can tip the staff say £30 at the beginning of the night and say "thanks in advance for making sure everybody in my party is looked after tonight!" ... but appreciate that might not be an option.
Do let the restaurant know that nobody.... NOBODY apart from you two will be allowed to put drinks on your tab. It's sickeningly common that people try that one.
Alternative; have you sent out a wedding gift list? Somebody could sponsor the fizz on arrival or wine on tables perhaps?