Hi @otis2611, I'm sorry Otis is going through this. No fun at all and can be very painful. I can only tell you of my own experience with my bulldog, Finn. He had a partial tail amputation a year or so ago b/c of frequent tail pocket infections too. The vet would have him on a combo course of antibiotics plus a steroid like prednisone, the infection would clear up beautifully, and then come back fiercely again.
how many doses has he had of the cephalexin? he really should take it on a full tummy otherwise if he continues to get sick, you will need to discuss an alternative medication for him.
Also, may I ask why you are against having the surgery to remove the tail? this has been pretty beneficial for many bulldogs, including Finn. I know @Libra926 also had great success with her boy Vegas.