It's just substituting gravitational redshift for cosmological redshift; one can also substitute kinematic redshift for cosmological redshift. The redder bits are running slower, which one sees from spectral lines that are reliably generated by quantum processes. There are reasons for preferring cosmological redshift, and in particular it's because whatever is going on with the distribution and motion of galaxies, it's doing an excellent excellent imitation of a big (~70%) stable energy-density smoothly at every point.

The reverse of your question is interesting: a quasar host galaxy or a Seyfert galaxy isolated deep within a supervoid should be much less redshifted than similar cluster and filament galaxies (one would have to "weigh" the galaxies, and so would lean on angle-diameter-distance and other geometrical relations, relative redshift of hydrogen-alpha (and millimeter CO and 21cm neutral hydrogen spin-flip etc.) at the galactic limbs and in globular clusters). The side-effect of Timescapes running clocks much slower in filaments is that they run much faster in supervoids. Many known voids have small numbers of galaxies in them (the emptiest of voids have something like a tenth of the radio-generating matter compared to large galaxy clusters, so they're nothing like "empty", which in turn drives the focus to the evolution of Timescape's "void fraction") so looking for faster-running galaxies in voids is not an impossible observational test.

Going forward again, the late-time Integrated Sachs-Wolfe structure would be anisotropic, with strong nonlinear differences between ISW radiation through filaments (because of the significant gravitational backreaction central to Timescapes) and ISW radiation through voids. Coarsely, that's not what we see (nonlinear effects are subdominant) and is an obvious target of study of the Timescapes proponents.

https://en.wikipedia.org/wiki/Sachs%E2%80%93Wolfe_effect#Lat...