It fails where you say possible truths are a subset of the contingent ones. Possibility encapsulates contingency.

It is false to say 'possible A' claims 'A or not A'.
It is false to say 'A or not A' is contingent.

'A or not A' is tautologous, necessarily true.

I resume my idea with other words:

"A" is necessarily true iff "not-A -> (B and not-B)"
"A" is possible true iff "A or not-A" is true
"A" is empirically true iff "A" is true

What fails in these definitions?

1."A" is necessarily true iff "not-A -> (B and not-B)", is invalid.
It fails when "A" is necessarily true if "not-A -> (B and not-B)".

(not-A -> (B and not-B)) iff A, is a theorem.
(not-A -> (B and not-B)) iff necessary A, is not a theorem.

2. "A" is possible true iff "A or not-A" is true
A or -A, is true for all values of A. Possible A is false when a is A contradiction.

3. "A" is empirically true iff "A" is true.
False, 'B or not-B' is true and 'B or not-B' is not empirical.

They are (informal) definitions, so their true is established a priori.
I want to say that a "unnecessary" conclusion can be "invalid" in two ways ("possibly" and "impossibly" true). MP inference is a valid scheme (necessary), AC is invalid but "possible" and MT asymmetric (to infer "A") is "impossible" to be true.
If something is not necessary, then it is contingent; if contingent, it can be possible and impossible. If it possible, also it can be "real" or "empirically" true.