It's not really axioms themselves that are nonconstructive it's the combination of axioms which cause issues with constructive interpretations.
LEM collapses a Cartesian closed category (even distributive categories) to be nonconstructive.
If you have a substructural logic (so a closed monoidal category) or more radically cointuitionistic logic (like Set^op) you can have constructive interpretations for versions of the law of excluded middle.
You can kind of think of the situation as being like exceptions or continuations can let you "observe the evaluation order of the interpreter." So you can either weaken identities (substructural logic) or ban some constructs out right.
It's not really quite an issue of there being a constructive interpretation or not. It's more an issue of there being constructive interpretations satisfying all the logical identities we want.