A trademark does not have to be registered to be enforceable. Registration is usually preferable because it provides certain litigation advantages, but in a case like this those advantages are totally unnecessary. MSU clearly has priority in the trademark, has been using the trademark for decades, and can easily bring the case in federal court, so there really isn't much else the registration would provide. Further, since Mizzou is in the SEC, I don't think they can claim they were unaware of MSU's prior use.
However, even if the mark were registered, MSU still has to prove a likelihood of confusion among consumers regarding the source, sponsorship, or origin of Mizzou's services, or some type of affiliation with MSU, or an approval by MSU of the use of the mark.
With respect to the stadium itself and ticket sells, Mizzou would say that no consumer is likely to be confused. If you are in their stadium, you will know that you are not in Starkville. If you buy tickets from Mizzou, you will know that the seats are not somehow sponsored by MSU. And their club area appears to be the polar opposite of our LFL, so that no one would think they had copied anything other than the name.
It is possible that MSU alums could be confused by advertisements to tickets to Mizzou games in the "Left Field Lounge Club" and think they were buying tickets associated with an MSU baseball event, but I don't know how likely that is.
OTOH, if they are going to promote their Left Field Lounge Club on any type of merchandise then there is a bigger problem. It all depends on how they intend to market it.
eta: MSU could also argue that "Left Field Lounge" is a famous enough mark that it deserves protection from dilution, and then it wouldn't matter if the mark were actually infringed. I'm not convinced that LFL would qualify as a famous mark, but in college baseball circles it just might.
I certainly think MSU should raise the issue with them, and I think they should change the name to just "Left Field Club" or something like that. If Cohen calls me, I'll do the work for free a substantial discount.