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?
It makes splendid sense.
One horsepower is 550 ft-lbs per second.
If your vehicle thruster produces one pound of thrust and the vehicle is not moving, it produces 0 horsepower. If it's moving at a rate of one foot per second, it has 1/550 horsepower. At 550 fps it's producing one horsepower, at 5500 fps it's producing ten horsepower.
Your understanding of the difference between "power" and "thrust" is a little weak.