I recommend you check the posts of contributors from Louisiana since everyone else's opinion will be based on experience in a different geographic area. With that said, if I were you I would look hard at the LSU varieties since they were developed in your area.

The LSU Purple would be a good choice since it bears 3 crops through the year. I would plant some Celeste trees also. If you go to the LSU ag center website they have good fig growing guide with recommended varieties for La. I like Celeste because it is what I ate growing up in La.