Motor runs great but generator does not produce electricity

Dying under load should not cause a problem normally.
 
When you say the capacitor goes bad each time, please explain how it does. Capacitor don't normally go bad usually unless they are under rated or very old. You can not install an electrolytic capacitor in place of a non-electrolytic type.
 
Please don't not attempt the procedure you describe on trying to recharge the capacitor. It is very dangerous if done. If the capacitor is an electrolytic it can explode if AC is applied.
 
With having been run just before failure there should enough residual magnetism in the rotor to generate the needed charging on it own.
 
From what I understand you may more problems than just a bad capacitor. In this case you need service info on the alternator to do proper testing. Please carry this generator to an experience technician that has generator troubleshooting experience; you are dealing lethal voltages.