Hello! I purchased a custom-fit Covercraft car cover for my A/C from
https://www.supercarcovers.com/ . I got one made from a material called NOAH. According to the research I did, it's a fairly good material, not the cheapest, not the most expensive, but it does pretty well at all the different things car covers need to do. I haven't had any problems with the cover scratching my car yet (although it's a fairly recent purchase), and I do take it off and put it on several times a week. I have no garage, and my yard has pine trees, so I have to protect my car from pine sap, cones, twigs, bark, and other assorted crap in addition to normal weather conditions.
My cover cost $230. It is kind of pricy, but I think it's a reasonable investment if you are looking to protect the appearance of your car, because it does cost a lot more to get the paint fixed or redone entirely.
As far as the battery thing goes, I've got no personal experience with leaving a car unused for that long a period of time, but it does sound like a reasonable precaution.
Good luck!