Why are Yukawa couplings regarded as fundamental constants if their values vary with scale? Why are Yukawa couplings regarded as fundamental constants if their values vary slowly with the energy scale (distance scale) at which they are measured?
This question is the same as why are quarks and lepton rest masses regarded as fundamental constants if their masses depend on the Yukawa couplings, which depend on the energy scale?
 A: All couplings in QFT are running (with scale), and are not constants, but are called such for historical reasons. Even the electromagnetic coupling runs, and its "constant value" is taken to be the infrared (long distance) value computed and fit. The strong gauge coupling $\alpha_s$ varies by a factor of two between 10 and 1000 GeV, (Fig. 9.5).
Luckily, Yukawas vary very slowly, cf. the black graph of Figure 8 here, faintly asymptotically free.
In HEP, it is customary to fix/tabulate the coupling at a given reference energy, and then adjust it in experimental fits according to the well-calculated (and experimentally verified) value at other energies.
The Yukawa couplings are normally fixed/defined at the EW SSB scale, roughly a quarter of a TeV, or at $m_h$, where the Higgs coupling defines the respective fermion's mass, $g_Yv$. It will be decades until one would be able to discern/confirm variations with scale of such Higgs couplings, so this isn't really a practical issue. The mass of the electron differs by a factor of half a million from the scale  to which the SM tracks its origin, but, hey, not even this matters that much.
