Ah, we all have our own unique LSAT nightmares, don't we? I understand what you mean. I have had too many experiences, not with the LSAT but with other exams, of considering staying up all night because I am too worried I won't wake up in the morning. What I would suggest is to, if possible, spend the entire week or two weeks before the exam going to sleep early and waking up at the time you would hope to wake up for the LSAT (if not even earlier). You are much less likely to sleep over your wake-up time if you've accustomed your body to waking up at that time.

Other than that, the standard setting multiple alarms, putting an alarm far from your bed are all things I do. If you live with roommates or family or have an early bird friend, ask them if they would consider giving you a call or knocking on your door if they don't hear from you by a certain time. Really, your body is very good for the most part at knowing when it can oversleep and miss a relatively inconsequential morning commitment and when it just needs to get up without fooling around. That is why I have had oversleeping problems before when I was really struggling with insomnia in the past but never once missed anything truly important. You'll have that nervous adrenaline that will make you a lot less likely to hit the snooze button or sleep through alarms.

TLDR; don't worry too much about it.