SMU and TCU are GREAT options that offer everything you both want. I may be a little biased because I’m from the area but truly I think these are great options based on what you’ve said.
ETA -
Your daughter wants “big time sports”, greek life and the South - check, check and check!
You believe she will benefit from a school that is smaller than a big state flagship and want easy airport access - check and check!
I will add that both school are in fantastic neighborhoods within dynamic cities with meaningful job and mentorship opportunities.
They are also expensive so important to make sure NPC shows that budget works. And there is definitely a “rich kid” element to the culture of both schools, which I think is common to many private schools.