I have one in my really small coop, here's how I do it.
Reinforce the poop board somehow. I would try taking yours out, and adding two 2x4's on the front end to hold it up. They will fly up to the poop board, then onto the roost. Just make sure that the poop board can support their weight.
My bantam coop is 4x5 (smaller than that on the inside, since it's insulated), and I managed a dropping board. My dropping board is 16 inches wide; how wide it yours?? I found mine too flimsy, so I stuck a scrap piece of T1-11 (1/2 plywood would work great) under the flimsy board. That way I can still slide the flimsy board out to hose off once a month or so (although I scrape it daily). I also have a little ramp, because the dropping board sticks out pretty far, and in a short space it's hard to make it up to the roost. Of course some just hop up on the dropping board, and then onto the roost; other's use the ramp.
In this picture, I don't have the plywood under the flimsy board (as I do now), but it'll give you an idea of how I've worked it. My next box (which is in there now) is elevated 16 inches so that I don't lose any floor space...
mine is about 20 inches right now, I'm thinking of cutting 4 inches off and using that 4 inch piece as my "gate" board at the doors to keep shavings inside. Removable. I'll try that and see if that seems more comfortable/roomy.