You're not getting Notre Dame unless you agree to play them late in the season when the weather has cooled down and you are well into Big 12 play. Texas tried to schedule them recently and we were unable to come to an agreement for those exact reasons. We wanted to play them BEFORE the conference slate started.
Not LSU or Ark because it's too much of a recruiting boon for them.
It should definitely be a non-conference game, though. No way do we give up Kyle for a conference game, especially the tards. Besides, Baylor has been assigned as their rival, so there you go.
Definitely Arkansas for a "neutral site" thing. And just rotating TCU, SMU, and North Texas or whatever other teams are up there and trying to schedule the away games for bigger stadiums works fine too, if you really just want a game in Dallas. Of course Dallas isn't really the point for Byrne, it's trying to come up with a game someone might sponsor. So that brings us back to Arkansas.
I actually like my recurring idea of establishing a 5 or 6 year series where teams of common background or interest play one another... land grant schools, schools with military histories, or something like that. Have half a dozen teams play each other over 5 years and give out a trophy and some scholarship money. I'm guessing you could get some sponsorship.
I'm pretty sure that LSU would want a home and neutral site to play. Home games for them and a neutral site like houston. I just don;t see LSU wanting to come to Kyle.