Capacitor to provide power to a circuit during a very small interruption
I would guess the camera takes 100-150 mA (180 mAh 3.7V battery, ~1 h runtime), and your LED will only be driven with about 20 mA, plus there should be a resistor in series with it.

I'd just power them both off the same battery if weight is a major issue, and if you're concerned about the charging voltage damaging the camera, add some cutoff switch (or just unplug it) while charging.