Twice now we’ve had one of our outside cats bitten by another animal and develop an infection. Swollen puss filled gross stuff. Cat didn’t want to be touched. They clean and drain, give it an antibiotic shot and send us home. Under $100 the first time, don’t know about the second because I wasn’t there and Covid made things weird with having to drop the cat off and leave it there for several hours.
I’ve seen the bills for the 24 hour animal ER when our dog was attacked when I was a kid, it was really expensive. And that was 20+ years ago. Avoid if at all possible. Go to the regular vet tomorrow.