I don't really understand what the problem is ..........
If You don't want the Load to be powered while Charging,
then just turn off the Power-Switch to the Load.
I would suspect that there is a significant amount of Resistance between
the Generator and the Super-Caps.
The Power Supply needs to be taken directly from
the Super-Caps terminals, and not from near the Generator.
The Super-Caps should be increasing in Voltage at a very slow and steady rate,
but by your description,
You appear to have the Charging-Voltage jumping up higher
than the Super-Cap Voltage.
Having the Power enabled again when starting to Charge,
is not an indicator of having a complete Charge,
You need a Volt-Meter, or LED-Charge-Indicator.
The following changes require a manual Push-Button-Switch to turn On.
.
.
.

If You don't want the Load to be powered while Charging,
then just turn off the Power-Switch to the Load.
I would suspect that there is a significant amount of Resistance between
the Generator and the Super-Caps.
The Power Supply needs to be taken directly from
the Super-Caps terminals, and not from near the Generator.
The Super-Caps should be increasing in Voltage at a very slow and steady rate,
but by your description,
You appear to have the Charging-Voltage jumping up higher
than the Super-Cap Voltage.
Having the Power enabled again when starting to Charge,
is not an indicator of having a complete Charge,
You need a Volt-Meter, or LED-Charge-Indicator.
The following changes require a manual Push-Button-Switch to turn On.
.
.
.
