I think it might be more important to point out that "lest" isn't used very frequently by the majority of native speaker (in BrE at least).
As far as your sentences are concerned:
1) It's not "eat much meal", it's "Alice doesn't eat much food" or "Alice doesn't eat much at each meal".
2) Alice is a girl's name so "he" is inappropriate.
3) If I were to use "lest" in this way, I would say "Alice doesn't each much at each meal lest it give her indigestion" or " ... lest she get indigestion".