I know you said that you want to keep the design of the set as original - which is fine, of course. But I'm also sure that you'd like to find the
cause of the trouble, even if you then choose not to implement the 'fix'. (Least, I would
).
So, with that thought in mind, is the cause due to unwanted feed-back within the wiring configuration / cct. design? I would liberally distribute 0.01 µF caps. around the set in the many appropriate and various places. Then, hopefully, the instability ceases. At that point, disconnect a total of half on those caps. and see if the instab. remains. You can continue like that, successively removing half of the remaining caps. until you find the critical one (or more!
).
Time-consuming, I realize: just depends on your 'need to know'; availability of time - plus patience and determination.
Al.