Does a non-constructive proof of bounds of a computable asymptotic complexity, with impossible fix, exist? Does there exist an algorithm, about which a non-constructive $\omega$-consistent theory $A$ can prove that it has time complexity $O(f(n))$ where $n$ is some univariate function of the input, but there is no $\omega$-consistent constructive theory which proves this bound, and there is a constructive $\omega$-consistent theory $B$, which proves that the algorithm terminates? If it is, is there such a case for a polynomial-time algorithm?
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.