OK, here are the test results:

I connected additional bypass capacitors across the cathode resistors
and screen grid, just as Tony suggested. Nothing changed. I also
checked the fixed tuning capacitors in the pi network, just as Kenneth
suggested. Nothing found.

OK, all the caps were fine. So what else was wrong?

Well, I noticed that in CW and TUNE modes, the peak RF output was less
than the peak audio output in LSB or USB. Strange. So I poked
around and determined that the diode-based balanced modulator was not
sufficiently “imbalanced” for CW and TUNE. I think this was a
combination of carbon resistor R402 having drifted higher in value
(it’s specified at 68K Ohms) and a bias supply voltage running a bit
low (should be +250 VDC, but runs at +230 VDC). So I soldered a 220K
Ohm resistor across R402 and now get full power in CW and TUNE because
diodes in the modulator are now sufficiently forward biased.

So why was low power a problem?

Well, the peak-power-at-plate-current-dip rule really only applies at
full output power when finals are running class B. When I was tuning
in CW or TUNE, and didn’t have full power, that rule didn’t apply and
it looked as if the finals weren’t neutralized. However, the apparent
neutralization problem didn’t show up when I tuned-up the SB-401 by
applying a pure tone to the mic input in LSB or USB.

-Dave, K3WQ