“The King of France is bald” is meaningful when France has a king and meaningless the rest of the time.

If you express this claim straightforwardly in first-order predicate logic, it can be either true or false depending on the structure you choose:

1. ∀x. KingOf(x, France) → Bald(x) — True, because there are no counterexamples
2. ∃x. KingOf(x, France) ∧ Bald(x) — False, because there is no satisfying value of x
3. Bald(KingOfFrance) — Erroneous because the universe does not contain an element "KingOfFrance"

If in France it is customary for the king to have his head shaved, then the first formalization is always true, and furthermore the original sentence has an ordinary interpretation which is still true when there is no king (though it is arguably better written as "The Kings of France are bald", to emphasize the scope of the claim, in that case).

The point I intend is that "meaningless the rest of the time" is not fundamental to all reasonable interpretations of the sentence, but a choice you made. (I'd also agree with gjm's comment that "contains a false assumption" is different from "meaningless". (And, yes, first-order predicate logic does not include that distinction.))