Please check out the followsing sentence.

(1) You had better not be absent from class lest you (should) not keep up with other classmates.

(2) You had better be not skip / miss class lest you (should) be left behind other classmates in class.
contiluo(1) You had better not be absent from class lest you (should) not keep up with other classmates.
(2) You had better be not skip / miss class lest you (should) be left behind other classmates in class.
The first one is okay.

In the second one, "skip" is an infinitive. You need to delete the "be."

Also, "in class" seems redundant, and should be deleted, in my opinion.

Don't be absent from class. ("absent" is an adjective) Don't be late.

Don't skip class. ("skip" is an infinitive - no "be")

To my ear, the use of "lest" is a bit dated, or old fashioned.
(2) You had better not skip class lest you be left bhhind other classmates.

What word can replace the word 'lest' ?
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contiluo(2) You had better not skip class lest you be left bhhind other classmates.
What word can replace the word 'lest' ?
I don't have a "single word replacement,"
but what I usually say and hear is:

You'd better not skip class or [else] you'll be left behind your classmates.

or [else] you may be left behind the others.

"Other classmates" is often used in cases like this, but there can be a small technical problem.

If you think of them as your classmates, then "other" could suggest that you are one of your classmates.

(It's not a big worry.)