I believe the TV contract is actually between the SEC and the various television partners they use. I would think that the TV partner likely has a decent amount of control over who plays/when, however, I also think the SEC has influence over those decisions, and the TV partners take their input into account.
I don't know how much the University themselves can actually do. They should be able to push the SEC to pressure their TV partners to be more equitable with the distribution of morning/afternoon kickoffs.
I think it's a fair request. LSU should have to play some day games at home. Tradition is nice and everything, but teams shouldn't be handed a competitive advantage, and fan-bases shouldn't be disadvantaged because of the desires of another school. It should be as equitable as possible.
You know this as well as anyone, but LSU has a VERY VERY VERY vocal fan-base that thinks nothing of complaining to the SEC about even the tiniest issues whenever they believe they're being treated unfairly. If Alabama and other Universities want to see gametime distribution change, they should probably adopt some of the techniques that LSU fans have used.
I usually prefer Auburn to have night games as well, but so far this year that's all we've had. I'm actually looking forward to a 2:30pm game, but I prefer those to be later in the year because it's just too hot in September (and sometimes in early October).
I actually think that LSU having a high-ranking and some recent success will lead to them getting pushed into some 2:30pm time slots, but I doubt the networks will try to force that on LSU at home. Ironically, if the shoe were on the other foot and Alabama only played night home games, LSU fans would be screaming bloody murder about favoritism.