Austin is a great town. You are definitely in the right place when it comes to deer hunting. The "Texas hill country" is hard to beat.
As related to your question, no, I have not heard of an outdoor range that prohibits FMJ bullets. I would think that the actual shooting range would offer you the answer to your question but, even then, if that's their policy, then why even ask? lol I would just go to the next range...and there are several in the Austin area.
Zachary