I don't know where you are living, but if it gets hot in the summer that fan will be running alot and the battery won't have time to charge. Perhaps if you had a panel big enough, but that would be getting costly.
How far is the run to your coop? Couldn't you run some buried wire (in conduit)...
I had a couple of chicks that had some pasty butt...and we just pulled it off it looked to be blocking. In all actuality, I don't think it was pasty butt, since pasty butt when the birds sit in the poop and it forms a crust (often from shipping). I think it is just that the baby chicks poo...