Don’t apply ED if you have run the NPC and it comes up with an amount you are not willing to pay. Rejecting an ED contract because you don’t like the amount of aid they give you even though it is commensurate with your EFC is not what the ED contract means.
Not sure what sort of aid you will get at 200k (I read 400k is the cut off of no aid expected), but various online calculators and the FASFA should give you an idea.
With a daughter who wants to go to medical school and being in state for UT Austin with presumably an auto admit, that seems to be a no brainer. I am sure your daughter would rather you shoulder some of her med school debt, rather than pay for undergrad and put the rest on her. Your post seems a little bit like prestige chasing, and that is something you need to figure out if you are willing to pay for. Of all the schools my son applied to, the only one we were willing to pay (and prestige chase :)) full freight for was MIT. Ended up on the waitlist. He is at our in state option, Georgia Tech, as an engineer - which like your in state option is excellent!