I have a similar set up.
I put down something for them to scratch in; leaves, pine needles, straw. What ever is available and/or cheep. And I usually lump it in a big pile in their run and spike it with BOSS and let them kick it all over.
I'll rake it up every 4-6 weeks when it's looking tired and put it all into the compost bin.
But each morning I quickly go through the house (which has sand on the floor) with a kitty litter scoop and scoop up the droppings and toss them into the compost bin. I also walk around their run and scoop up the larger droppings and put those into the composter too. I think my run stay's pretty clean that way. And if you do it daily or every other day, it only takes a few minutes. I should also confess, I only have three hens so we're not talking about a huge amount of droppings.
Not sure that routine would scale if you had say 20 hens or more.