Thanks to all of you for your suggestions.
I did call SW this morning and they were very understanding, but however, they do not have a cut and dry policy concerning this.
To the previous poster, my husband does not have to come in direct contact with a pet, just being in the proximity is enough for a asthmatic attack, but as he posted this is not common.
They have red flagged our reservation, we are to confirm this at the ticket counter and again at the gate. I was informed that we would most likely pre-board and be asked to sit at the front of the plane then if there happens to be a pet on our flight, they will be seated as far away as possible from us. She asked if the separation on the plane would be enough, and if not one of the parties would be asked to be bumped to another flight. So I guess we'll find out the answer to this when we get on the plane. Wish us luck.