Hello calzap,
[...] it decodes to a minimum of 500 uF. Would like to replace it with one rated 250 VAC. Haven’t been able to find one of that capacitance at any voltage. Any opinion on whether a 540-648 uF capacitor, which is common, would be OK to use?
Such a huge value is usually used as starting capacitor, which is disconnected a few seconds after motor start. Do you actually need a start or a run capacitor ?
Here is a cheapy :
500 µF 250 VAC start capacitor. Do not use as run capacitor.
its not critical at all.
For a start capacitor, you're right. But if calzap's issue is about the run capacitor, I would not bet on that : main and aux coils currents should be about the same value and the actual run capacitor value is therefore important.
[...] Motor started to turn over slowly, then overload protector in controller tripped. Although there is no burnt smell, my guess is that the insulation on some of the start winding wires had melted, and the conductors are touching. Will do a teardown to check.
Hm, the defective underrated capacitor seems to have spawn additionnal damage

It would be interesting to know if a non-destructive teardown is sufficient to locate the failure.
In some cases, you don't see or smell anything, but newertheless you have a single shorted turn or maybe as you guessed, an insulation issue.
As suggested above, making a ringing test or measuring the motor's current can sometimes be more reliable than visual/smell inspection. If you have an identical replacement motor, you may also measure/compare the coil's inductance.
I imagine that a megohmmeter(*) used for insulation testing has already its' place in your toolbox, so this is an easy test to do.
(*) Top notch megohmmeter for industrial environment :
-
Fluke 1507-
Fluke 1587 FC