JAX has a/c, it just isn't noticeable if they keep the front door open or if you end up sitting next to the kitchen where an eliminator isn't removing kitchen heat and it comes into the dining room.

You could make a reservation and say that being comfortable is a concern right up front. I was there on an unusual night when there was a table for 27 expected that prevented them from moving us.

Or if you like dining al fresco you could sit outside.