The FP2 is known to be very thirsty, indeed. Some things that might play a role in your case: The (likely) simultaneous double load of streaming and HD (?) video playback and (perhaps) encryption of the device.
One easy thing to check regarding reboots: The battery bay was a tiny bit too long on some FP2s, so the battery might lose contact from sudden movements. Try if you can force such an interruption when the back of the FP2 is open, by trying to move the battery away from the contacts at its top end. If you think that’s the case in your FP2, you might fold some slim paper and bolster up the space between the battery’s bottom end and the end of the battery bay.