Has anyone tried using a motor starting capacitor of the same value and voltage rating with an HID ballast? If so, what were the results? If there's a reason not to, please tell me.

Starting capacitors, probably not, but running capacitors will do, given it has the same capacitance and the same (or higher) voltage rating. I think starting capacitors get very hot with extended operation (possibly boiling and exploding), I hear that they are bipolar electrolytic but I could be wrong.

Agreed. Motor start capacitors are not designed to be powered for very long.

Motor RUN capacitors on the other hand double as HID ballast capacitors just fine, if capacitance-matched and rated at the same voltage or higher. As a matter of fact, this contraption of mine uses an old dehumidifier fan motor capacitor, works flawlessly!

Starting capacitors are allowed to be powered only for few seconds at a time, with minutes cool down dwell time inbetween.
Otherwise they severely overheat...
Often they are in the form of bipolar electrolytic, with rather high dissipation factor, so creating a lot of heat when connected. They can swallow quite a chunk of it (the few seconds for the motor to start), but for longer time their temperature will rise so much the electrolyte starts to boil and that state becomes their instant death...
These are designed to have high capacitance to start larger motor in a small, and inexpensive format, but the price to pay is the very limited time they can be connected at a time, really just start the motor and once it is running, the capacitor is supposed to be disconnected.

Thank you gentlemen for the very helpful info. I appreciate it!

One advice - you better verify the capacitance if you are reusing older capacitors. Metallized film capacitors tend to lose capacitance with the age, rather significantly!