"Velcro strip." ... find a piece of velcro large enough to cover the hole and cut the barbed side to fit around the hole and the soft side to cover the hole.
Duct Tape works wonders in a situation like this. It isn't a permanent fix. It will, however be a great 'quick fix' to get you through the snow, rain, etc. until you can replace the back.
Where I live we have Tap Plastics. A store where they sell all kinds of plastic. they could cut you a little thin piece that would cover the battery compartment. a little drop of silicone at each corner and you are good to go.