Welcome Cats and Chicks! Bedding, as you can see, is a hot topic around here. I'm fairly new myself so I'm no expert but I can tell you what I've experienced. I too have a small coop (see my profile picture) and use shavings. My girls come out during the day but my husband has been in the hospital for about a week now (he's fine & should be home soon) so I haven't been doing much besides feeding them lately. This is causing them to become a little anti-social and they are staying in the coop more often. With this, I have found that the coop doesn't stay clean quite as long. However, I have also found that if I clean the clumps and what not out of the coop every other day or so, things are much better and I actually go through less waste because I don't have as much to clean out because it isn't stepped on and dragged all over the coop. I picked up a small beach rake & shovel (minus the pail) at a Dollar Tree store and use those to pick up the clumps. The coop smells better, the girls are happier and it's easier for me than if I wait and let it pile up and mush together. Actually, they get a little testy if I wait more than 3 days and they start showing signs of not being happy, such as knocking their roosting bar to the floor, or pooping in their food container.
You may find that it's probably not necessarily your hens that are making it smell bad and cause the flies. It's probably the treats you throw in. I have a small feeder in my coop as well and don't really have a problem with waste on the floor. If you are putting kitchen scraps or other treats in there, they are probably rooting for them, which is great, but not catching them all before they start to decay, hence the fly problem. Is it possible to get a treat ball or some container that you can put the treats in? This may help as you can take them out if they aren't eaten by the end of the day. A treat ball (or other puzzle like container) is great because it keeps them busy, which keeps them happy.