My old Chihuahua's eye bulges from a bad tooth. What can I do?

I would add that I think it's unlikely for a tooth root abscess to cause the eye to actually pop out (Proptose) however without definitive treatment the problem (the bulging and the infection) are going to get worse. Having that pocket of infection in the gum will likely cause the bacteria to translocate to the bloodstream, where it could cause other problems, including with the heart and the liver.

I would consider seeking another opinion, perhaps of a veterinary dentist. I have anesthetized many, many older dogs for tooth removal, and my personal opinion is that removing diseased teeth is necessary. If Enzo has blood work prior to surgery, and possibly even a chest x-ray to check his heart and lungs, in the hands of a qualified veterinarian who is confident with anesthesia and dentistry he will likely do fine.

If you don't have the tooth removed, your only other alternative is using antibiotics in a "pulse" fashion, such as 3 weeks on antibiotics, followed by 1 week off. Clindamycin is sometimes used in this fashion, but again, results are suboptimal and the infection within the root pocket will never completely resolve - it may improve, but it won't resolve.