Hey OP, thanks for taking him to see a doctor. Your local vet should charge A LOT less for treatment than an emergency clinic would. My advice as someone who works at a general practice is to be completely honest about what you can afford. There’s a wide spectrum of care from full anesthesia and wound cleaning to more conservative management with medications and the vet can discuss what the best options are. I really feel for you and the stress you must be feeling right now.
Having teeth extracted is unlikely to affect how he eats. He is much more uncomfortable having teeth affected by infection or periodontal disease. Cats don’t do much chewing in the first place. I have known toothless cats that still eat dry food.