Sounds and looks like the battery could be a little loose and lose contact.
Try to fixate it against the contacts with a piece of cardboard or folded paper on the opposing side of the battery compartment.
I suspect the battery loss is not real, but yet another software issue. A device can never determine the exact amount of charge left in a battery.
Maybe the random reboots come just as surprising to the Fairphone as it is for us users the first couple of hundred times. Logs may stay incomplete, calculation tables might not be closed, so the device software needs to assume something.