I also performed the "just push it" experiment proposed by WarpTech. When the wedge is moving I saw a slight decrease in RL for the three tests I did. In the chart, Light blue is stationary (RL-29.54dB), while dark blue is when the wedge is moving (RL-29.10dB).
I usually avoid to comment on EM waves aspects because it's not a lot in my field, but given the order of magnitude of wavelength, dimensions of cavity, Q, and accelerations involved by a "just push it", I'd think it's just plain impossible to detect a resonance shift due to something pertaining to "redshift". Actually it would be quite a flabbergasting experimental result in itself to show a rigid device (RF generator rigidly attached to rigid cavity) with similar parameters can be used as an accelerometer (of such sensitivity) as it seems to contradict Maxwell equations. I'm not speaking here of a resonance or phase shift induced by some mechanical deformation of cavity (or RF feed) induced by the acceleration, but a direct (small but significant) acceleration effect on the EM waves in an idealized perfectly rigid system (of similar parameters), really 
I know of laser gyros but this is not really same wavelength, is it ? Anyone with more EM fu might prove me wrong on paper. Unless, I would consider this to be the result of mechanical deformation (of the cavity, or of the RF feed).
Anyhow, I have a quick and cheap experimental proposition that also (in idealized system with Maxwell only) should return null (no resonance or phase shift) but could conceivably show interesting slight deviation in reality : see how the readings react to the presence of a (strong) permanent magnet probing the system around, the system being static (fixed). This could also be related to the "just push it" effect : a moving (not necessarily accelerating) system in geomagnetic field might pick up some induced currents and/or change DC magnetic conditions around, even if it sounds a bit of a stretch. At least if the permanent magnet probing returns null we know such explanation to the push effect is excluded, and if it doesn't return null then that adds another unintuitive effect to investigate. Thanks.
If the light's energy is really equivalent to the energy transferred to the cavity during acceleration then maybe such a device, able to pull so much energy from light, if it works in reverse, could be quite an efficient accelerometer? That is if it's resisting acceleration can transfer as much energy to the light.
I am assuming that by F.dx there is a force in the forward direction and the cavity moves in the forward direction then an equivalent energy is drained from the light inside. So the reverse being the same force but being accelerated in the opposite direction then transfers energy to the light?
I am still unsure if there is any propulsion of course, but even if there "is propulsion" I am also unsure if it is just between the light and the cavity or there is some 3rd medium that allows the conservation of momentum.
With a 3rd medium, then even if the cavity is sitting still it could be accelerating this 3rd medium and doing work. If the cavity is only interacting with the light then if the cavity doesn't accelerate then no work is being done and so no energy should be lost. So which is it?
One might say, that momentum must be conserved so there must be a 3rd medium. That sounds good, but I can also think back to a photon recycling thruster where a force is acting on the mirrors but the mirrors are not accelerating so no energy is being lost until the mirrors start accelerating. There doesn't seem to be a 3rd medium needed for a force to exist on the photon recycling thruster with 2 separate mirrors but at least that system seems to obey the conservation of momentum. This of course brings me back to the effect of light entering a dielectric and becoming more massive, inducing a back reaction on the dielectric, and the more massive light being more effective at transferring energy to a reflector.
So maybe, if there is an accelerating dielectric of free space and the light is pushing on it, but the acceleration of gravity is greater, and the system is set up so the movement of the dielectric of free space is against the force from the light. As a result there should be some transferring of energy from the dielectric to the light. I guess this would be if the EM drive is pointed down? This is assuming some 3rd medium form of propellant. I am assuming something similar to what is in this paper here:
https://scholar.google.com/scholar?cluster=17356997040342833631&hl=en&as_sdt=0,48 ?
DOES THE QUANTUM VACUUM FALL NEAR THE EARTH?
Tom Ostoma and Mike Trushyk
I am not really sure how seriously to take this line of thought but I am just throwing it out there.
Speaking of permanent magnets is interesting. I know stationary charge shouldn't be able to observe stationary magnetic fields but the current in motion inside the cavity should be able to observe them, similar to a hall effect probe maybe. I would imagine getting a permanent magnet close would change the observed modes inside the cavity because of the change in the path of the current. There would probably be at least some resistance to a change in the magnetic field observed. Similar to dragging aluminum near a magnet and moving the magnet or dropping a powerful magnet over a plate of aluminum.
I have questions if the cavity would be able to keep resonance if a powerful magnetic field was near by, and am curious what would happen.