.05 amps out of a $10 7amp hour battery means you get 5 or 6 days of reserve power, and that is if cai uses 12v, but it actually uses 9v, so you'll probably get more. The battery should do fine at cold temps, but if this is an issue you can use a $100 battery instead of a $10 battery that would also give you a couple months of power without recharging. Batteries work fine in the cold, you lose some cold cranking amps, but at trickle levels it won't make much difference. So if you got snow on your solar panel, it would certainly melt or slide off at some point prior to the battery going dead.
I just look at your situation and I see, what, 30,000 feet of cabling?. This would be very expensive and labor intensive to run. Are you going to bury it? Are you going to run it in conduit? Animals like to eat insulation and if it is exposed to sunlight you have to worry about the insulation falling apart with UV damage. Unless you spend the money to run all of that wire in conduit, my suspicion is that you are going to have significantly more maintenance issues with wire than solar/battery.