Good then, and yes it's not simple. In any frame, inertial or not, having a non-zero x-axis speed component wrt lab i.e rocket frame. Thus in both the belt frame and K frame. My reading of your 2nd, treadmill scenario in #30, which is what we have been dealing with, is that observer at rest in K has an x-axis velocity such that lab, and mass-on-scales on moving belt, have equal and opposite x-axis velocities seen in K. Which remains true even when y velocity becomes appreciable. But there will be corresponding equal and opposite x-component decelerations. Good, there is some progress now. Time-rate-of-change of y velocity yields in turn time-rate-of-change of overall time dilation factor, which in turn means uniform motion in lab frame is non-uniform i.e. accelerated in say K frame, or any other inertial frame having an x-component of velocity wrt lab/rocket frame. You have that basically right. I specifically chose a dilute gas for the reasons given there. In the case of a solid or liquid, there is essentially equipartition of molecular/atomic/lattice thermal energy between KE and PE since one there has a system of excited oscillators. Which is different to the case of a gas at typically encountered pressures. The observed net mass increase will be the same in all cases, but with a different split as to the detailed contributions. I do suggest to ponder that example in #15 some more - it's simple way of tying relativistic mass and Lorentz contraction together to show how that γ² factor arises naturally. Groan - more punishment for me.