A basket works well on a boat but IMHO the visibility thing pops up again. You need to see your way around on deck.
Also, there are those large barrel style line and rod 'baskets' that would probably work well although I haven't tried them myself they are popular on boats.
They sell spring-frame popup baskets (mesh hampers, vinyl yard barrels) that fold completely flat and pop up to similar dimensions as the barrels used on boats but lightweight and removable. I would imagine you could attach it to snaps on the stern deck and pop it up when fishing, fold it flat when not. Might be worth a try.