CRI
Registered Chooser
General anaesthetic is worrying in any circumstance for any animal, but I don't think any vet would do dental surgery on a cat with just local anaesthetic, cats are quite strong, have multiple weapons, and even the most good-natured can be prone to panic when their mouth is being fucked with. One of the reasons cats have dental problems is that it can be pretty much impossible to ever brush their teeth, even the softest easily-handled cat can go a bit 'what in the holy fuck do you think you're doing? - Deploy Weapons' when you try to use a finger brush in their mouth.
He doesn't have bad breath, nor any trouble eating (obviously if he had trouble eating it would make this urgent!) - if I hadn't seen for myself the state of his gums and teeth, I would have found it hard to believe. He spends a lot of time right up close to me and doesn't have halitosis.
I must confess, I've never tried to brush a cat's teeth, but I can imagine the outcome! Hope the op does go well (and not too hard on the pocket.)
Anaesthetics are particularly risky for cavies partly because they can't be given IV (veins are too small) so the vet has to use gas and it's harder to moderate the dosage. The other thing is they can't be "out" for long as they have to eat at least every couple hours to keep the gut moving continuously, or it can just pack up. And, they aren't good with stress of any kind - likely to just shut down.
But, luckily they are pretty easy to immobilise in a carefully wrapped towel so all but the most invasive procedures can be done under local. I had a fantastic rodentologist in South Wales who could trim and file a full mouth of teeth in about 10 minutes - pig hardly had a chance to squeak! One of mine had to have that every fortnight or so, and if he'd had to be anaesthetised each time, he'd have probably only survived a couple of rounds. With the right instruments and techniques, it's not actually that difficult to do cavy dental work, but frankly, I don't think most vets can be arsed. Too many still see rodents as "disposable" pets.
(I'm going to shut up now before I launch into a rant about a certain vet college and certain "exotics specialist" who seemed almost terrified of touching Griff and every visit took a cursory look in his mouth and said, "oh, I can't see the back teeth as his mouth is full of food," - doh, their mouths are ALWAYS full of food, so you've got to have the confidence and tools to get in there and "get involved" or you're wasting your time and risking the animal's life, how the fuck did this person ever get their job . . . )
(Off to take blood pressure medication.)