First off, I would point out that the mainstream story has changed over time. The current view is not that it's a Doppler shift that's causing the redshift (and time dilation). Instead they call it a "cosmological redshift" and attribute it to a "stretching" of space time. Now does the spacetime stretching idea imply that there should be time dilation? I don't know. But let's say it does. Then it may not be too hard to connect that with a tired light mechanism too. The waves in a wave train need not be stretched as a result of expansion, just stretched through a loss of energy. Incidentally, the fate of the lost photon energy in the mainstream view is considered problematic in mainstream discussions.

You're focusing on the fact that no one has come up with a tired light mechanism yet that is free of problems. I would agree we don't have the mechanism yet (but would add that my own mechanism, not fully worked out, could avoid most of those problems. See my recent paper in Apeiron). The common error people make (including the authors of the paper you mention) is to equate the tired light concept with a specific mechanism (eg, Zwicky, Compton) and then conclude that the SN data prove the expansion case. Tired light can still be used without a well-defined mechanism to get a better handle on cosmology. The precise mechanism can come later. We can be sure the actual tired light mechanism, once it is found, will show time dilation too. Time dilation actually helps the tired light case with respect to the Tolman surface brightness test.