Finally the problem is identified and solved! Here is the procedure I've been through.
So the symptom was that sourcing below -350 Volts the resonant high-voltage converter became loud and the output voltage dropped. I have changed the output filter capacitors (orange ones) and the dummy load at the output. It has not solved anything. In fact one of the ceramic caps was blown. After many hours spent on an incorrect direction, I have put a 120kohm dummy load on both rails of the converter. It has sourced the 10mAmps perfectly. Converter was locked out as a problem-source.
Output stage was the next step. The voltage dropped on the Q3 & Q4 seemed good with an output voltage of -50 V. Each of them dropped abound 400 V. When the output was raised lower than -170 V the voltages dropped on Q3 & 4 were fluctuating from 250 to 350 V. Below -350 V the converter started the loud oscillation and the voltages dropped on the FETs were zero. I have changed Q3, Q4, Q10 and Q19 but with no success.
After all this I have stared monitoring the -150V rail on C19. It was -170 volts actually. Going down vith the source voltage this rail was decreasing down to around -400 volts!! CR4 is a high voltage diode with the purpose to separate the high voltage rail and the -150 V rail. It seemed the diode is not acting this way. I have changed it to a big ugly one and everything is fine.
AFAIK there were not any 237 related problems here like this one. I hope it will help somebody one day.
A fellow member help me a lot solving this issue. Thanks z01z!
If anybody needs an MTP2P50 FET or MJE350 or MJE340: it is available here at the local dealer just PM me. They are geniue.
Next step is calibration. Also I would like to pimp my 237, with changing the high voltage wires to nice, flexible silicone ones, replace the 150kohm resistors near the supply to higher power rated ones, etc. Thanks for all your help!