I think back when the HP204C was designed, HP would have been extremely happy with an amplitude change less then 1% over time. Minimizing distortion would have been a higher priority then stability. It will probably stabilize if you leave it running long enough, but back in the 60's, the common meters were moving coil and the sort of changes you are talking about are hardly noticeable on a moving coil meter. I think you are suffering from a condition "Fluke189ivitis" which is worry triggered by the extra resolution of your excellent multimeter.
Lucky you do not have an 8 digit meter - the HP204C would look even worse then.
If HP were using resistors with a 200ppm/C temp coefficient, then if a resistor warms up 10 degC, its value can change by 0.2% - that is the variation of a single component. I am trying to remember how they stabilize the amplitude of that oscillator - I think it is a zener and JFET circuit, but it may be a light bulb. Both will have a temperature coefficient. If it is a light bulb, then as the temperature in the HP204C case rises, then amplitude will fall.
Richard