Portable Generator Battery

I have a Champion 3100W RV ready Portable Generator that I bought last Spring for extended dry camping excursions. I'm not sure how to store it for the brutal winters we have here in the North. I called Champion and was on hold for an hour. I emailed over a week ago and still haven't received an answer so hopefully someone here can help. I keep it in our unheated shed and now that winter is approaching, should I keep the supplied trickle charger that comes with it plugged in all winter or should I make sure the battery is fully charged and just remove the Pos or Neg cable from it's post and call it good? The shed does have electricity. Thanks.