Did some maths and also "real" tests today. The traditional linearizing technique with a coil in series with the antenna comes to its limits at such high frequencies as used in the Theremax (>800kHz). Out of that there are big differences in the pitch antenna design of the original Theremax and the PT-03, which has a 60cm long and 4mm thick thread rod. So the Paia method with a relatively small capacitor in series works better as has been proved in practice.
In order to be sure I lowered the frequency of both pitch oscillators to about 480kHz by soldering temporarily 220pF capacitors in parallel to the 100pF in the tank circuits. This allowed to improve the linearity by replacing the 5pF capacitor by two 4.7mH coils in series (as expected). But the amplitude of both oscillators became so week that the quality of the audio signal was unacceptable.
Conclusion:
My first task, making this PT-03 work correctly (as a Theremax) is accomplished. I had to fix one of the coils (a replacement was not available), to exchange a 390k resistor with a 4.7M (production fault) and to replace the LM339N (wave shaping did not work and there was no pitch cv). Out of that I tin-plated most of the circuit board because of very poor etching quality (lots of hairline cracks). As written above, I do not at all recommend the purchase of this bad Theremax rip-off.
My second task, doing some tests for a future project (still a secret), could be done after the device worked correctly and was successful.
My further idea, improving the linearity (which is already not bad at all by default, compared to the B3 and the Silicon Chip) could not be realized without important changes on the oscillators design. So I decided to postpone that until I'll have a real Theremax for myself (my birthday is soon...) instead of tinkering a friend's instrument.
I will meet him next Monday/Tuesday in southern Germany, give him back the repaired PT-03 and test another of my new circuits in his Etherwave Pro (but this will hopefully be the subject of a new posting next week).