Propellantless drives violate conservation of linear momentum, but not necessarily conservation of energy.
A violation of conservation of momentum in one reference frame would violate conservation of energy in another reference frame. Although if you are throwing out conservation laws I guess you might as well throe out special relativity too.
I'm not "throwing out conservation laws"; the one possibility I pulled from my backside could conserve both linear momentum and energy.
Let's consider two inertial reference frames, one before and one after some acceleration with a propellantless drive. In the first, the violation of conservation of momentum appears to increase the total energy, and decrease in the second. This is special relativity. For special relativity to hold, in actuality, the total energy must be conserved instead.
One solution to that problem is to couple the drive, somehow, to a larger system, so that for the entire system both energy and linear momentum is conserved.
If we consider warp drives like Alcubierre drive, they only seem to violate conservation of linear momentum. They do not, because the (let's say) "anomalous velocity" of the warp drive does not increase the linear momentum; it only exists as long as the drive is "active", and is actually due to the space contracting and expanding with the drive. (If we observed such a drive, we'd probably see an optical distortion caused by such a drive, but had no other "hints" that it wasn't violating conservation of momentum or energy.)
Now, I have not examined all propellantless drives to find out if they only seem to violate conservation of linear momentum but actually do not (either because of spacetime distortion or by coupling forming a larger system where both linear momentum and energy is conserved), or if there are other solutions to the conservation of energy in different inertial reference frames while violating conservation of linear momentum, for example via quantum-scale effects. Because of this, I prefer to err on the optimistic side, and assumed other ways might be possible. If I had preferred to err on the side assuming all of important physics is already known, I would have written something like "Propellantless drives only seem to violate conservation of linear momentum. They do have to conserve both linear momentum and energy, just like for example the Alcubierre drive does, to not violate physical laws."
I did expressly write I think the one discussed in this thread is extremely unlikely to be a real effect, but it has more to do with the physics models ("laws") that have survived the tests of new measurements –– including special and general relativity ––
refining or extending previous models rather than replace them.