Certainly the Pye AM15 Westminster and W20AM/FM Whitehall could do 100% modulation or better; problem was, Pye back then had two different microphone-inserts with different frequency-responses/outputs for AM and FM. Put a FM mic on an AM radio and it would sound 'thin'.
Sticking with the CODAR-style circuit then experimenting with outboard audio-processors sounds a sensible path; it certainly makes it a lot easier to change things round to see which gives the results you like.
There was a bit of experimentation with 'interesting' modulation schemes in the US in the 1950s:
https://www.rfcafe.com/references/ra...ruary-1950.htm
and the attached PDF which shows the 'negative cycle loading' thing with a high-power diode on the secondary of the modulator transformer with a potentiometer to set the point at which the diode loads the secondary on negative voltage-excursions.