Poor baby, it does sound like a dental abscess.
I dealt with this with my boy Moose, he had a rotten tooth that caused the abscess. I would encourage you to go to a vet that has lots of dental experience, and pursue the most aggressive treatment possible if you want a true cure.
The usual treatment is to find the problem tooth (it might need to be extracted…usually easy if the tooth is rotten), then to clean out the abscess. It could also be that he has a molar spur that irritated the gums and caused it (so the spurs would just be trimmed). Rabbit abscesses have a very thick pus, so the vet will need to manually clean it out. Sometimes they also pack them with something, but not always. Then they will do a course of injectable antibiotics that you give at home (usually penicillin given sub-cutaneously), plus pain meds for the first few days while the extraction is healing.
It’s awesome that he likes critical care on his own, as he will likely eat it readily after the procedure as well.
After the full course of antibiotics, there is usually a follow up exam (with Moose he was put under again) to check to make sure the abscess is gone and that no other teeth have become a problem. Sometimes follow up cleanings of the abcsess are needed.
When Moose was diagnosed with his, I did ask about whether it was possible to treat with just antibiotics. The vet said that in some cases where the bunny is not well enough or strong enough to undergo the surgery and anesthesia, they will put the bun on antibiotic shots indefinitely to prevent the abscess from getting larger. But it will never cure it or even really shrink it, and even injected penicillin can cause problems over the long term. So really that would only be the case for a bun that was very ill and poorly in other ways.
Moose had a bunch of other problems, but his initial abscess did resolve completely after the first procedure.
My vet has also neutered buns as old as 9, so if Tutu is otherwise in good health he should be able to handle the procedure! I encourage you to get him scheduled for the procedure as soon as possible, with a confident vet.
Sorry you are having to go through this.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.