No, the new idea is a massive photon. There's strong limits on that from astronomical observations, and a stronger one from Special Relativity when cast in a modern isometry group theory form. In that form, the Poincare Group is the isometry group of the unremovable (flat space) background, and the Poincare Group has exactly one free parameter, "c", which corresponds to the speed of a massless particle. In this form, we assume that light is massless, and look for experimental evidence supporting that assumption (there is a fair amount; we have lab studies showing that the m_photon cannot be more than about 10^-17 eV/c^2 and there are even more stringent limits from observational cosmology).
A nonzero photon mass makes a mess of particle physics at high energies. The Standard Model falls apart due to loss of gauge invariance, and QED becomes really obviously non-renormalizable near limits we have already tested. So this would be a big thing.
(Note that if you bite the bullet and take a nonzero photon mass (as McCullogh says in his paper at page 3: "Normally, of course, photons are not supposed to have inertial mass in this way, but here this is assumed") you can probably get a non-constant photon speed too, along the lines of neutrino oscillation. But you can always write down non-physical theories that conflict with frequently and precisely tested areas of physics...)
None of the above departs from the symmetries of Special Relativity, and one subgroup of those (invariance under spatial translation) is what implies the (local) conservation of (linear) momentum per Noether's (first) theorem.
So there's a really big looming question about why the Standard Model (which has the Poincare Group baked into it) works as well as it does everywhere but in an EmDrive or a single space probe, and resorting to special shapes of objects on scales much larger than that of atoms further conflicts with Poincare invariance.
Finally, Unruh radiation is a difference in particle count and particle energy measured by differently accelerated observers. The cosmological horizon, when it formed, produced an acceleration between observers then and observers in the future. Unruh radiation from that is pretty uncontroversial. However the problem is that the acceleration is pretty small, so the temperature will be much lower than that of the CMB. Also, you'd expect anisotropies based on Earth's (and its surrounding Local Group's) peculiar motions relative to the horizon. Why isn't there a dipole anisotropy similar to the one we see in the CMB, and if there is, by how much does it offset the inertial argument in McCulloch's paper?