You have chickens - you have a garden - no reason why you should not compost! In fact it's a waste if you don't! 1/5 acre is not too small! You can make a compost bin out of almost anything - wood, concrete blocks, or just a circle made of 3 T-posts surrounded by wire. It doesn't have to take up a lot of space, and it doesn't have to be complicated!
The key to it not smelling bad or attracting flies is:
1. Keep it contained instead of spread out in a big pile, so it will cook. You need it pretty compact for it to generate heat, which discourages insects and kills their larva and eggs.
2. Allow air to get in, and keep it moist but not soaking - the wire-circle style works well for this when you are starting out. If you live in an area that gets a lot of rain, you might need to have a cover for it.
3. Make layers of 1 amount "green" (nitrogen: like chicken manure, vegetable and fruit scraps, fresh lawn clippings) then 2-3 amounts of "brown" (carbon: like fallen leaves, shredded paper or cardboard, dried lawn clippings, wood chips, straw, twigs.)
What I would do to start out in a smallish yard: Drive in 4 T-posts to make a 3x3 square near your coop, and wrap them around with poultry wire. Every time you clean the coop dump it in there, then cover with twice as much dead leaves and twigs, office paper from your shredder, ripped-up pizza or
amazon boxes, crumpled pieces of newspaper. Same for kitchen scraps, every gallon you dump, cover with 2 gallons of "brown" stuff.
Grass clippings are awesome to add anytime, they heat up really quickly and get the whole pile cooking. If you get a ton of rain, put a cover (tarp, plywood) over to cover the compost container.
As the container heats up and decomposes, it reduces in size, so by next gardening season you'll have some good, healthy soil to add to your garden beds.
Composting doesn't have to be hard or complicated, and it doesn't have to be smelly or noxious to your neighbors, or take up a lot of space!