Low power device with high current peak
This circuit might do what you want. (It's just an idea and not finished).  

It has 2 charging rails for the supercap. If the supercap is empty, it slowly charges through R5 to prevent overload. If the supercap voltage is above 2.5V, it charges the supercap directly. (Of course you still need to fine-tune the circuit to your needs)







simulate this circuit &ndash; Schematic created using CircuitLab