6+ days to remove & reattach power seems like plenty.
skyjumper has been using a supercap with boost regulator to provide time to complete data writes to an SD card. Designed his card to separate out the parts that really needed power for the writing, let the rest go unpowered.
Has some diodes to keep it from backfeeding into the charging circuit, and a resistor is needed to current surge at initial power-on. Resistor also slows charge up time, your tradeoff to make there.
Make sure you get low ESR super-caps.