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?
>The cone allows Unruh radiation of a certain size at the large end but only a smaller wavelength at the other end. So the inertia of photons inside the cavity must change as they bounce back and forth. And to conserve momentum, this must generate a thrust.
The new idea introduced is the quantization of inertia at small accelerations. As far as I understand it, from one end of the cone to the other there isn't a smooth change of inertia. This depends on the idea of Unruh radiation, and the reason the article gives for quantization of inertia is that as accelerations get very small, Unruh radiation wavelength becomes larger than the observable universe, forcing unruh radiation to take whole-value wavelengths (quantization). Again as far as I understand it, the inertia of photons on one end of the cone takes a different quantized value of inertia than the photons on the other end. So the thrust isn't a violation of conservation of momentum. The thrust is necessary to not violate conservation of momentum.