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Getting the discriminator proplerly centered on the 4.5 MHz carrier helps reduce its sensitivity to amplitude variation. As described above, the dropout of the video carrier is lowering the amplitude of the sound carrier. Getting the discriminator exactly centered gives you the best chance of minimum sensitivity to the buzz.
I imagine the reason it deteriorates with time as the set warms up is that the tuning is drifting. You may find that simply retouching the discriminator secondary is enough to reduce the buzz after the set warms up. If that turns out to be the case, changing the type of capacitors used in the discriminator to a zero temp coefficient or negative temp coefficient capacitor may help keep it stable. It's OK to use ceramic caps in these tuned circuits if you find the zero or negative temp coefficient. Otherwise mica is probably better.
Make sure all stages of the sound IF are very nicely aligned. If you have a sweep generator, go for the most beautiful symmetric discriminator transfer function you can get, with the zero crossing right on 4.5 MHz.
You may even have to tweek the video IF as well, since too much rolloff in the video IF can alter the symmetry of the audio demodulation response. (Even if the 4.5 MHz IF is beautifully aligned, asymmetry can be introduced by the stages prior to the 4.5 MHz IF.)
All that being said, there are some sets (for example, Philco Predictas) where I also cannot completely get rid of the buzz when there is white lettering on the screen from a DVD player or computer generated graphics in a broadcast signal.
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