That's still constructive. This can be a confusion because there's actually two kinds of contradiction proofs:

  to show ¬ P, assume P then derive a contradiction (false, ⊥)

i.e., ¬ P := P -> ⊥. This is actually just fine and constructive, it's just how you prove a negation.

OTOH, there's another kind of contradiction proof:

  to show P, assume ¬ P, then derive a contradiction (false, ⊥)

Written with function notation, this becomes P <-> (¬ P -> ⊥)

But if we unfold the definition of negation, we have that

  P <-> ((P -> ⊥) -> ⊥)

The left to right direction holds classically and constructive, but the right to left direction uses double negation, which can is equivalent to the law of the excluded middle, AKA classical reasoning.