No one is going to want to hear this but the design of the PA output filter is probably fatally flawed and it is likely due to board layout issues.
Pic #2 shows the path 80 meter RF follows through all three switching relays in order to reach the filter. It is a very novel and inventive concept to use three relays to switch both ends of 4 filters. Most manufacturers would require 8 separate relays to do this job. And for a very good reason.
In order for a filter to provide the required 50+ dB of attenuation, the filter itself must have 60 dB or more of input-to-output isolation. If it does not then the unwanted products simply bypass the filter and go directly to the output. That appears to be exactly what is happening in the filter complex of this unit.
There is no easy fix for this. No amount of on board filtering and tweaking is going to improve harmonic suppression until the path around the filters is blocked. I suspect that path is via the relays themselves but board layout often is critical as well in such cases. My guess is the only solution would be either extensive external filtering or a carefully laid out daughter card to replace the existing filters and relays.