For less than the cost of the batteries you could rent a ditch-witch, drop in a run of #10UF cable, set a small six circuit 240v MLO panel, set a two-pole 20A CB in the house end and a couple of 15A CBs for 120v circuits on the shed end.

I would have to run the numbers to confirm the wire size but 15A on #10 at 200' each way sounds about right. Simple enough to do the calculation but it is late.

Two 15A circuits would give you all the lights you could want and a circuit for tools, a small space heater in the winter, a fan in the summer. A lot simpler and reliable than messing with batteries that will need replacement, solar panels, chargers and all the rest.

You can rig a solar setup on the house where you can more easily mess with it.