I closed the level-control loop and the PLL. After a few more small changes it works.
I skipped the second part of output amplifier (was unstable) and the first part seems to work good enough. I also skipped amplifiers after VCOs and added attenuator after 2-2.5GHz pin attenuator (too much gain). Variable attenuators were too sensitive to tuning voltage (at certain frequency and tuning voltage) so I introduced some imbalance as workaround. Other changes are listed in changes_rev1 text file in the project zip.
Recently I had chance to access a good spectrum analyzer for a while so I checked the results. The output signal looks bad (spurs, jitter), partially (mostly?) because I didn't even shield anything. Also I'm not 100% sure if everything is stable and not overdriven. The main tone is about 12dBm, it drops to about 8dBm close to 7GHz but in the remaining part of frequency range the output level is surprisingly flat (maybe about 2dB flatness below 6.7GHz). See the video in "files" (note 10dB attenuator was installed).