Propositional truncation of excluded middle
The only way to prove ∥ X ∥ is to prove X (unless you admit some other axiom).
So, assuming P is a proposition, there is no way to prove ∥ ((A) ⊎ (¬ A)) ∥ if you cannot prove ((A) ⊎ (¬ A)).

It is not a correct intuition that undecidable propositions are either true or false, just we do not know about it.

A formal system S with an undecidable proposition P can always be extended either in S'=(S,p:P), or S''=(S,p:¬ P) where p is taken as a new axiom. Both S' and S'' are then consistent formal systems, provided S is.
If P were already true or false in S, then either S' or S'' would be inconsistent.

A formal system is meant to be interpreted, and can usually be interpreted in different ways. Undecidable propositions in a formal system are propositions that can be true in some interpretations, and false in other interpretations.
A proposition is said to be true (resp. false) within a formal system when it is true (resp. false) in all possible interpretations. So there are lots of propositions that are neither true nor false within the formal system.

The confusion comes from the different meanings for 'true': the truth within the formal system usually does not match exactly the truth in the interpretation (or intuition) that you have in mind when you are working with the formal system...