Pardon the deconstruction, but @Emilio 's answer only addresses the straightforward dimensional analysis reclamation part of your question. Its deeply wrong-minded premise, however,
But I think that a decay constant should have a dimension of [T]−1, where [T] is the dimension of time.
merits redress. As you can easily check in your textbooks, a decay constant is virtually never proportional to a decay rate, Γ (which has the inverse time dimensions you are looking at here). It features in amplitudes, normally squared to produce rates.
In the decay of a charged pion to leptons, say, muon and its neutrino,
\Gamma_\pi \propto ~G_F^2 ~ f_\pi^2 ~~ m_\mu^2 m_\pi ~~.
So it hardly makes sense to associate this constant, merely measured from pion decay, to an inverse time. It is an auxiliary parameter firmly residing within QFT and never exiting it.
In case you were interested in the "point" of the constant itself, it is but
the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action.
In turn, it is, broadly, the scale at which a (current) quark-antiquark pair "morphs" into this pseudoscalar meson and connects it to the vacuum—so it can devolve to a virtual W which then decays to the leptons. I have never met anybody who could visualize this process in terms of times.