I have experience taking lessons both at AM and FADS. In my opinion, the major difference between these two organizations is the emphasis on technique. While there have been many champions come out of AM, I think that the emphasis at AM is on "social" dancing, and technique takes a back seat. It takes a special student and a special teacher to succeed outside the AM world. On the other hand, FADS seems to put a lot of emphasis on technique, as witnessed by the many US and world champions that FADS has produced. And, sometimes it seems that the "social" aspect of dancing gets lost at FADS.
In my opinion, I believe that this difference comes from the "founders" of these organizations. Arthur Murray was renowned for teaching the current "fad" dancing of his day, and even had a string of TV shows. His emphasis was to get everyone out on the floor and enjoy dancing. Quite the opposite, Fred Astaire was a stickler for technique. He insisted that all his partners perform his choreography perfectly without any errors. He held himself and his partners to a high level of technical achievement. After the Cold War was over, there were a lot of immigrants to the US from the Eastern Block countries, and many were excellent dancers and teachers who found their home at FADS.
Both organizations have found success in the dance world. AW is, I believe, the second oldest franchise operation (KFC is the oldest). Others may make comments about AW, but you cannot deny their success. FADS the same way.
Go to an organized competition and you will usually find students from both AW and FADS there, and usually there are more FADS students.
The problem with your AW franchisee and outside competitions comes right down to "money". If you take a teacher off the floor, that teacher will not be able to conduct lessons, and the studio will lose money while the teacher is gone. Thus, the reluctance on the part of the owner. Have you set down with your teacher and talked to him? Have you talked to the owner? If so, and the owner still will not support your desire, then, yes, find another studio. And, another studio could be another AM or FADS studio, in addition to the choice of independents.