Does a non-constructive proof of bounds of a computable asymptotic complexity, with impossible fix, exist?
No (assuming $f$ is computable).  By $ω$-consistency, the $O(f(n))$ bound is correct, so some $c f(n)$ bound is correct, and we can add $c f(n)$ bound (which is $Π^0_1$) as an axiom to a constructive theory.