I have seen this sort of thing (not on one of the QRP Labs kits, though) and found either a bad capacitor (heating and changing capacitance) or a bad inductor (heating and changing inductance). One was a coupling capacitor and another was in the LPF. The inductors that I have seen change were in the LPF.
If you are seeing a larger power output when cool or after a cooling period, then the part heating (bad part or resistive solder joint) will be my guess. Maybe trying some "cooling spray" or air directed at the area to see if the power comes up again. Use a microscope, if possible, and look for cracked capacitors.
Measuring the power before the LPF will produce unreliable results due to the square(ish) nature of the signal before the filter (I have likened class E to a "kick and coast" type of amplifier - one of the reasons that they have high efficiency and low heatsink requirements). You can get a relative idea that there is power there, but the comparison from stage to stage will only be relative and not a power referenced to the 50 ohm output. Possibly using an oscilloscope to trace the signal might show where distortion (it is going to be pretty ugly before the LPF anyway) is introduced. The QCX uses a class E final. Look up some class E waveforms or the expected waveform - it is far from a sine wave until after the LPF. shows the theory - practical circuits have more going on and will be messier.
---------
Paul KE7HR