I do know extreme cold takes it's toll on batteries just like extreme heat, but heat's worse. Never went through batteries anywhere like I did in AZ!
AK probably qualifies for extreme cold, may just pull them and throw tham on a maintainer in the garage.
Running the pump dry does not remove enough or all of the water? My only experience with this is on my boat, the washdown/shower pump is the same type of jet pump and I've run that one dry and drain the inlet and outlet lines for years before winter layup without issue. What do those that do not antifreeze the fresh water system do about the pump?