One solution would be to get a garden. Yeah, right.
When you take that stuff out you might have two situations. The really nice compost may be segregated out at the bottom with the stuff not quite finished at the top. If that is the situation you can use the compost in your yard, just spread it out. You can try filling holes and low spots. Or use it in landscaping beds. Leave the stuff that has not composted in the coop/run to finish composting. Or move it to a compost pile.
If you are not comfortable spreading the stuff that comes out on your lawn or in landscaping beds, pile it up somewhere and let it finish composting.
If you are in the US, contact your local master gardeners. It is highly likely someone will clean it out for you just for the chance to haul it away. Or advertise it for free on Craigslist or whatever the Craigslist equivalent is in your country as long as thy are willing to clean it out and haul it away.