I'm going to go out on a limb and say the reason we didn't/couldn't use ULA is because it's the same as trying to use UL...any time we use the U (university of), it has to include the city designation, even if it's an abbreviation. I think LA fits the whole "athletic branding" initiative while keeping it legal, as it's just a shortened version of the Louisiana athletic nickname and wouldn't require the Lafayette due to not including "University of." But I could see some media pushback on LA because that may or may not be used by any number of Los Angeles franchises, I don't know for sure. Thinking a bit outside the box, what about LARC or LRC for "Louisiana Ragin Cajuns?" Similar to how the Dodgers use "LAD," the Lakers use "LAL," etc. I don't know how much I love them, but those are different and they get rid of the need to use two L's.
EDIT: As I was typing, Jay confirmed that ULA would violate the law as I thought.