Those are all great choices, and I am a huge fan of all four (my grandfather was a Vandy alum, and my brother in law is a GT grad).
I would definitely include both Rice and GT. My guess is that Tulane’s quota might fill up early because they provide free room and board to all ROTC scholarship students. On the other hand, Vanderbilt gives $6,000 toward room and board to ROTC scholarship students, so its quota might fill quickly too.
So, take your pick between Vandy and Tulane, but include both Rice and GT. My personal favorite is Rice, and it is probably the least well known which might keep it from filling up.