How to do high switched current from small solar
Just imaging your high power circuitry as one "device" and your low power stuff as another.

Now, branch the power from the super capacitor to the 2 "devices". The low power branch can be permanently connected, but for the high power branch use a p-channel mosfet to switch the power on and off. You can use an NPN transistor to pull the mosfet gate down.

Now the only other thing you'll have to be careful of, is to not parasitically power the high power systems through the IO. Basically you'll have to make sure that all the data lines are kept low. If any of them are high, power will flow into the high power side through the protection diodes.