I thought that was implied by the description of the EM drive of "X kilowatt for Y newton". But I guess the effect could indeed depend on your speed relative to some special reference frame, like that of the microwave background.

If that's the case, then we can do something like the Michelson–Morley experiment, to check whether the effect is different depending on the direction the Earth is moving in its orbit.

Good point. Or if it is emitting em waves they would get red shifted as it sped up. (Not sure if that’s ruled out)