I finally decided to do what PE1HZG has done and I am very pleased with the result.
Removing the onboard 404 MHz LO resulted in destroying the oscillator, I must improve my de-soldering skills. The new LO was built by DF9NP, 404MHz output at +5dBm with a connector for a 10 MHz external reference, it came ready to use in a nice sheet-metal enclosure. A short pigtail cable was soldered to the oscillator input pad and Resistor R1 on the transverter board was removed. The pigtail connects to the LO via a small hole in the front of the original transverter enclosure. A bigger box is planned for the future.
After this modification I am happy to report that the transverter finally is stable.