how about adding a small solar panel to keep the batteries charged? I've even seen strings of LED lights that come complete with battery and panel at Costco.
I see a few things I'll just mention, but experimentation is always the solution.
That is a mighty nice big perch you gave them out in the run, I can see why they want to stay outside. Also, I found my chickens don't like hay, but prefer pine chips. I also ONLY use the chips in their nesting...