Welcome! My name is***** am a UC Davis graduate, and currently, a Medical Director of a veterinary hospital.
I am sorry to hear about this concern for Ginger.
In my experience, I have not heard of any tooth root abscesses involving a carnassial tooth that resolved without surgery. I guess the only exception is when a tooth was so rotten, it fell out of the mouth, and the subsequent oral antibiotics remedied the problem. That happened only once, and was not ideal for the patient, due to the severity of additional dental
disease that was present.
Homeopathic options? Zero, and I strongly feel that is true.
No, that should not be the cost to extract a carnassial tooth and do treatment.
Unless there is something else going on, I would definitely get a second opinion from another local vet. My prediction would be that someone can address the mouth for far less cost.
As a educated guess, the cost should likely be $1000-2000 maximum, and that would possibly include extraction of any other diseased teeth.