Bi-Intuitionistic Logic adds to Intuitionistic Logic a binary connective $←$ known as co-implication or subtraction. A weak negation $\sim A$ is defined for Bi-Intuitionistic Logic as $\top ← A$; Bi-Intuitionistic Logic proves $\sim A \lor A$ for any wff $A$.
It is known that a variety of basic results in set theory and arithmetic imply the full Classical Law of the Excluded Middle, including the well-ordering of the naturals. Supposing someone were to use Bi-Intuitionistic Logic as a basis for a theory like $PA$ or $ZF(C)$, would results like $\sim A \lor A$ lead to a collapse of the two negations down to just Classical negation? I know that in pretty much any theory of arithmetic, the Classical LEM is equivalent to the well-ordering of the naturals. Is a weaker LEM strong enough to prove a result that would turn the whole thing Classical?
Modulo the caveat that you already have to make some concessions from $\mathsf{ZFC}$ to adapt it to intuitionistic logic, no. Bi-intuitionistic logic is conservative over intuitionistic logic, so while it may have use in elucidating the symmetry that classical logic collapses, it will not induce that collapse in and of itself.