Using your current clamp for both power and signal would result in a significant degradation in the accuracy of your measurements. The high turn-count of the secondary on the CT requires thin wire, producing a relatively high output resistance. Applying the dynamic load of the Arduino would change the load on the CT output, causing inaccuracy. While inductive energy harvesting is definitely possible, you wouldn't want it to be from your primary sensing device.
As for using electrolytic supercapacitors in your application, be aware that safely charging/discharging them is best done with a dedicated IC for low-power backup applications. Such controllers include the LTC3225, the LTC4425, and the LTC3226. However, these controllers would require having a low-voltage supply available normally such as a 5V USB power brick/"wall wart". For higher voltage inputs, you could use a Li-Ion battery and the LTC4079.
Implementing inductive energy harvesting could be done with the LTC3588-1 and a second CT. You will need to ensure that the CT output voltage will be appropriate to drive the energy harvester at both low and high measured currents. I am uncertain how the combined energy harvesting and supercapacitor implementations would interact and they might not be easily combined.