You're not reading this correctly. It's questioning the assumption that the Supernova are of the same brightness no matter the age of the star. Because the supernovaing stars are on average older now then they were in the early universe, there's a systemic bias in the data making it seem like there is red shift when there is not. No new physics required outside of a new understanding of how Type Ia supernova work.

Do the current models of dark energy not have it changing over time? Or at least allow for the possibility?

The idea that fundamentally, physical law does not change over time, is either a metaphysical or methodological assumption (or a bit of both), but it is certainly not an empirically-derived result, and it clearly does not rule out manifestations of those fundamental laws changing over time.

The issue is much more prosaic than the fundamental laws of physics changing over time. On average, very distant supernovae have a different composition than nearby supernovae. As the universe ages, more light element atoms are fused into heavier element atoms and expelled into space. That material is incorporated into the next generation of stars, so stars begin their lives with less and less hydrogen, as the universe ages. If that influences the light curves of supernovae, then it greatly complicates the use of supernovae as standard candles for distance measurement.

That's what is weird about this: from the paper, "Unlike population age, we find no significant correlation between host metallicity and Hubble residual". So it correlates with age, but not metallicity!