Whatever mass you're kicking out of the back of the vehicle is accelerated relative to the vehicle. No matter how fast the vehicle is going, if the mass-flow-rate and the delta-v is the same, the thrust produced is the same and the energy cost is the same.
You do not conserve energy in your case. KE = 1/2 mv². The higher the velocity, the higher the cost per incremental change in velocity. Since, P = F * v, your rocket would become much more powerful the faster it goes if it had a constant force. Does it make sense to you that an object that hasn’t changed its mass becomes more powerful the faster it goes when it is only powered by a constant power source?
Oh, and before you mention it, an accelerating body is, by definition, not in an inertial reference frame. So your Galilean transformation arguments do not apply.