They don't need much inside! If you don't want poop all over the top of that wooden nest box, I suggest putting an angled roof on it. Given you have plenty of height, you might even make a storage unit with angled roof/lid. Maybe keep your 10 gallon metal can with lid and bail feed storage in there.
I saw the nesting boxes with angled roofs. Am going to work in nestung area. Does make it easy to clean.