I it was soemthing easy to try but I'd call it the same as you. Lots of harmonics
in square waves and little filtering.
At frequencies below 20MHz the filters at the amp output are the only filter.
Not my favored way, either.
If there were band pass filters at the amp input this would not be an issue
nor would spurs be an issue.
One way and likely there are many:
An improvement is to move the keying (CW_TX) to the product detector/ TX modulator
and move the BFO to the edge of the pass band and key that. The easy way is
remove R105 and run a wire from the junction of C1/R104 under the board to
a replacement R105 at the modulator junction C222-1 and the transformer T7.
Then both the 12 and 45mhz filters can clean it up. That would be the wire changes.
Doing that requires a code change. Actually any change that comes immediately
to mind requires that. The code change would remove the mix of VFO switching
to only the BFO and the signal CW-key would drive the Balanced modulator. That
works well as It has been done on many other radios. Code is easier than solder.