@Keith Bryant hey, this is actually one of the nicer properties of supercapacitor-based systems compared to lithium — supercaps handle simultaneous charge and discharge pretty naturally since there's no chemistry to destabilize. The Power/Program Kit is designed around that kind of always-on use case, so in principle your solar-in/load-out setup should work without needing an external charge management circuit sitting in between.
That said, the one thing I'd watch with a small solar panel specifically is input voltage regulation. Supercaps will happily accept whatever you throw at them up to their rated voltage, so if your panel has any open-circuit voltage spikes (which small unregulated panels definitely can), you want to make sure the input side is clean. If your panel has a built-in MPPT controller or you're running it through even a basic voltage regulator, you're probably fine. If it's a raw panel direct to the kit, I'd put a simple buck converter or clamp circuit upstream just to be safe.
Worth flagging this directly to Moamen in the Dev Hub too — he'd know the exact input voltage tolerance on the kit and whether there's any onboard protection circuit handling that already.