Well, I'm a bit more nuanced in my "choice" of rival.
When it gets down to it, it's mostly about which one of these particular teams is the most potentially damaging team to face. Which one might pull out a win. Which means that even A&M and Ole Miss, for split seconds, were rivals needing to be beat (and that they have been, one way or another).
As it stands right now, I'd say it's a LSU >Auburn > Tennessee ranking.
But I still hate them all, just for slightly different reasons.
Tennessee, because I remember my dad really disliking them, and during my adolescence, they were really good, up to and through the Peyton years. Mid 90's to mid 00's weren't a great time, so I'm continuing to be happy with every year this winning streak continues. Looking to see #11 in a row in person (first time to actually go to a TSIO game).
Auburn, is mostly because they're the fans I know most of. I've met and interacted with more Auburn fans than any other college fandom. I have relatives that are Auburn fans. Best case scenario: There's not much sports talk or just very slight ribbing about each other. Maybe even mutual respect over "this years teams" and how they look.
But most of the time, they're assholes. Very similar to those asshole younger brothers or cousins you might have had growing up. Yeah, you can be civil, but you can tell they have the courage to be pricks. Having the downside of seemingly every Auburn victory comes on the wings of some disastrous event (Camback, Kick Six, Probation leading to "Fear the Thumb"), doesn't help.
LSU is the combination between nouveau riche and jilted ex-girlfriend. Bit of a younger brother in them, too, during the Les years. Always trying to build their team in the same way as Saban did it, but with diminishing returns.
Could throw in Florida, as there's been plenty of "big games" between the 2 over the last few decades, but Florida's been in such a bad downturn, it's lost the luster it had in 2008/2009 (and the 90's). But besides a bad Prothro joke by pxleyes, there's never been a huge amount of rival hate.