Is there an exception to the prohibition against ending a sentence with “ ’s ” at work here?

The ’s can be used as a contraction representing a weak, unstressed word that is not pronounced. It allegedly cannot occur in sentence final position.

• She is not ready, but he is.
• She’s not ready, but he is.
• She’s not ready, but *he’s.

The last one is not grammatical there.

Similarly, here the last one is not grammatical:

• He has not started yet, but she has.
• He’s not started yet, but she has.
• He’s not started yet, but *she’s.

This answer claims this is because you cannot end a sentence with a weak form. It backs up its point with reasonably scholarly documentation.

However, this does not seem to be invariably true.

• Shall we go to the movies? Yes, let us go.
• Shall we go to the movies? Yes, let’s go.
• Shall we go to the movies? Yes, let’s.

Unlike the third example in the first two sets, here in this case the third example is indeed grammatical, despite ending with a weak form.

Is this an exception to the rule, or is there another rule at work here?

It's worth noting that they're contracting different things:
He's/She's = He is/She is (not he has/she has, as you use it in the second set of examples)
Let's = let us.

So "he's" is a contraction of the form N+V while "let's" is V+N.

Other N+V contractions can't end sentences:"
"Is there a dog here?" "*Yes, there's"
"Is that a mammoth?" "*Yes, it's" "Will you take me to the store?" "Yes, I'll" "Are you santa?" "Yes, I'm"

But contractions with the verb first, you can:
"Should I jump over this alligator?" "No, don't!"
"Will you go to the moon with me?" "No, I won't"

And possessive 's can also end sentences:
"Why is that cat staring at me?" "Don't worry man, it's Tom's"

So it seems like the only contractions that can't come at the end of a sentence are those which the contracted portion is a verb!

Hope that helps!

The idiomatic expressions "let us . . ." / "let's . . ."

meaning roughly

"here's an idea; I strongly suggest that we . . ."

are, I think, always interchangeable on grammatical grounds (though some older Church leaders might consider "Let's pray" over-informal, and often sentences like "Let us go to the cinema tonight" would sound far too stilted).

However, "let us . . ." meaning "allow us to" is never rendered "let's". Idioms often seem to have their own rules.