Well, on a basic level you're free to use whichever vet you like - practises can't stop you leaving or being registered in multiple places.
But the vet world is small and any new vet is going to want your pets' medical history and details of your previous practise. Any future insurance claim will require this as well and if they contact the previous surgery and find out you have an unpaid bill, they may refuse to cover you.
I suspect it's not uncommon for clients to find themselves blacklisted if they have a history of refusing to pay - ultimately vets are private businesses and, unless it's discriminatory, can easily refuse to take you as a client.
If you don't pay your bill then be prepared for it to be taken go collections and ultimately to court. It would probably be much cheaper to pay upfront and then complain afterwards imo.
The main problem you'll have is "satisfactory" is very much down to individual interpretation and the vet will likely argue they had your animals' interest at heart which is why they did X and not Y, for example.
Whenever any of my animals have gone for surgery I've had to sign a form to say I give them permission to operate and that I understand that costs may vary depending on what's discovered while under anaesthetic.
So I guess it depends what has happened and what your complaint is about as to whether you'll be able to reduce your bill.