Kinetic energy is 1/2 MV2. In this equation, kinetic energy is a property of mass and velocity squared.

If we look at the velocity squared term, squaring a number not only impacts the scalar, but also the vector. Squaring means a plus or minus vector direction, ends up the same, in terms of kinetic energy. Kinetic energy is not relative to direction and therefore it is not relative to any given reference, since the vectors connected to red and blue shifts are normalized by the square term.

As an experiment, we can start in empty space with zero gravity and use X fuel to generate a final velocity for our rocket. This kinetic energy will be based on an energy balance; X fuel. We can launch the first experiment moving away from the earth and then the second experiment toward the earth. Based on energy conservation, since we used the same fuel both in all experiments, both experiment will have the same kinetic energy. However, special relativity calculations will show differences in motion coming and going, relative the earth. The slowing or speeding up of space-time, due to relative reference, does not alter fuel usage, energy conservation and therefore the kinetic. These only add an energy mask to the true kinetic energy.

The lack of dependency on kinetic enemy on reference, can also be inferred from dimensional analysis. If you look at the units of kinetic energy, it is M d2/t2. It is mass times, double the units of space-time (d,t). This tells us kinetic energy is not a function of one space-time reference, but rather of two references. The real kinetic energy, based on energy conservation, is not impacted by the red or blue shift seen by different references, who may see this motion coming or going.

Kinetic energy needs to be measured against a standard that is not connected to space-time; C-ground state. At the speed of light space-time breaks down, so one reference is possible for a standard. In space-time we need to use two reference and then take an average of sorts.

I often wonder if we are properly conserving energy, when we look out at space and see the Doppler shift using one space-time reference as the standard. The result of a red shift only approach; from earth reference, would take away energy. The result will be hidden energy that needs to explained in others ways; dark energy.