After rereading the entire thread, I think that the problem is a stuck centrifugal switch.
The centrifugal switch is supposed to take the capacitor and start winding out of the circuit as soon as the motor rpm gets close to full speed.
This should happen in about 2 to 3 seconds. The fact that the capacitor is failing after 3 seconds indicates that it is still in the circuit when it should not be.
A start capacitor cannot take continuous voltage for more than a few seconds. The continuous voltage is causing the capacitor to fail.
The start switch on that pump is difficult to fix. I would recommend that you run the pump to a local electric motor repair shop so that they can replace the switch and capacitor.
Email them first to inquire about the cost and timeframe. Include a picture of the motor label and a description of what you want done.
If the cost is too high, you might want to replace the motor or try to fix it yourself if you feel comfortable doing that.