I have been thinking about this for a while, especially Medina lake being deep and green it reminds me of being back in Washington when I fish it. It looks like a lake trout would love, I just don't know if there would be cool enough water with enough oxygen and food supply year round, but at 250 feet deep I would think so.

If it would have been possible, believe me, TPWD would have done it!
Back in the late 70's and 80's they were doing a lot of experimenting and they dumped several non-native species into our area lakes but the only thing that stuck was the redfish in the two power plant lakes in the area.
I know they tried Walleye in Medina but that kind of fizzled out.
Medina lacks nutrients that produce large amounts of bait fish and just about 0 vegetation.

I believe they survive (at least to an extent) in parts of the Brazos

I think they could make it in Wheeler Branch. I think it’s partly spring fed and deep.

Probably not cold enough!

Nope, upper water will get to warm and the deeper water won't have enough oxygen in summer.

https://texasfishingforum.com/forums/ubbthreads.php/topics/3757742/Re:_Canyon_Lake_Walleye

Stocking report for Canyon Lake

Walleye 1985 64,001 Fingerling
Walleye 1985 2,112 Fry
Walleye 1984 3,925,000 Fry
Walleye 1981 4,370,000
Walleye 1974 371,080
Walleye 1973 1,068,920
Walleye 1965 500,000