You know you may have to trap the kitty, the vets or shelters will usually rent you one cheaply, and you can put the food in and the cat will go in for it and the spring pops and the back drops down.They may want to give the cat shots etc which is a good thing .It probably is in a lot of pain hope you can help it .good luck
First, IF there is any possible way to get this cat medical attention, please do. It sounds pretty serious and if the cat can't eat, it can't last very long.
The Vet may be able to give you antibiotic to mix in with food, which will help. But it sounds like the tooth needs to come out and the socket cleaned before the infection will clear.
Second, IF there is no other help, try Colloidal Silver Water from the health food store. Mix a tablespoon into gravy style food and feed in a non-metal dish. Do this every feeding for a week or so.
I've seen this help humans get through some bad infections until they could get to the dentist. A friend who managed a feral colony also used it with good results on some untouchable cats. If nothing else, he would corner them and mist them from a sprayer, so they would lick it off and get dosed, or he could spray their wounds from afar.
It is far from the best solution, but may help get her through until you can tame her and get her to the Vet.