I have ADD, actually, so I also have had problems with concentration and "endurance" in the way you describe. I've found some ways to compensate for it, though. One thing I did when I was younger is to find a topic of interest and do it as long as possible - you won't have endurance for the "boring" stuff if you don't have endurance for the "fun" stuff.
For me I would usually read books that were 400+ pages (this may not seem like a big deal to most, but when your attention span is at most a few minutes...trust me). I was eventually able to work myself up to concentrating on the full book from start to finish.
Also, since multitasking is another thing I do to compensate, I would usually try and find ways to "multitask" one task. For you the "one task" is the LSAT, so it may help to do Logic for a certain period of time, then reading comprehension. Go back and forth between them a little bit longer than you feel you can handle (only a little bit). Each time, try to increase what your "little bit longer than you can handle" spot is. Keep at it until you can work through a whole section, and then a whole test, without feeling burned out.
That's actually what I'm doing right now
EDIT: I'd be careful with the caffeine. If you want to try that route, test it before the day of the test. Personally all caffeine does for me is make my heart feel like it's beating way too fast, but I still feel tired/foggy/unable to concentrate. One thing that does help, though, is keeping hydrated. (Not soda/juice/etc., just water.)