Love this idea.
In some cases, if you insulate the shed that can double as good sound absorption. Might be an option.
If you live somewhere cool, you could get by sometimes without air conditioning. It can be a problem like down where I am. The absolute quietest air conditioner you can get is a mini split. The heat exchanger goes outside, like a tiny version of central air. They are unbelievably quiet. A window unit will be a problem as it is pretty much the loudest air conditioner you can possibly get.
You got two batteries, 5 and 9. Likely a typo.
Solar might be an option too. In sunny places, 230 watt panels are last year's news and huge solar farms dump them for this years higher profit panels. Then you see massive stacks of panels that are not that old at habitat and resale stores for dirt cheap. Even one panel, a cheap charge controller and an old car battery would power everything in there no problem. Might be worth looking into if you don't have an easier option laying around.
Super interesting idea.