Welcome to the forum and so sorry u are going through this -
I too have a cat with bad bad teeth. She only has 4 left (canines) and they will need to be removed eventually. Vet thinks it's stomatitis in our case...but this is a general word as far as I know, as is dental disease. In our case, 'stomatitis' was determined by blood work which suggested an immune problem - and the lay conclusion is that she is allergic to her own teeth or rather the plaque that forms around her teeth. She has had about 10 teeth extracted thus far.
I can tell you that our cat gets regular abscesses on the 4 teeth that she has left and that it is painful. As the abscesses grow, she sleeps more and is less playful. (vet thinks that's pain). We do have been trying 'pulse' antibiotics and they do help for a short while. But they are a band-aid solution in our case.
I really don't know what i would do in your situation. 13 isn't young for a cat but many cats live to 20 years these days. The bad breath is a classic symptom of stomatitis, and the cat rubbing on the left side of the jaw may well indicate pain in that area.
If money is no object, then I might look for a feline dental surgeon and get an opinion there. It may well be that your cat needs additional extractions. And if that is the case, from recent personal knowledge about my own cat, the techniques for extraction by a vet surgeon are far superior to the techniques done by the average general vet. (Surgeons pull the harder to extract teeth via a cut they make in the gums, general vets pull from the top and hope the teeth don't break and when they do break they hack away at the remaining teeth).
If you do go that route, perhaps they might also have additional experience with anaesthetics.... and if they did and suggested a decent outcome, I think I'd take the risk.
One thing you didn't mention is the type of 'operation' the vet is considering.
Hope this experience was of some help and hope members have additional suggestions.