Great question. Poop management is an issue for all of us.
There is a whole lot of information on composting over on this forum's sister gardening site at the bottom of the screen, TheEasyGarden. We could really use more people over there, asking questions, giving advice and sharing experiences. Composting this stuff can be real simple or it can get complicated.
You can make this as complicated as you wish, looking at how you garden (square foot method, raised beds, container, turn a patch over in the spring, do you mulch), when you clean out the coop or plant your garden, what kind of plants you grow (some burn from the poop easier. Watch out for squash and cucumber type plants especially), the materials you use for your litter, the poop to litter ratio, what other things you add to your compost (grass clippings, garden wastes, kitchen wastes, pure poop from a droppings board), your climate, and probably a few other factors. Or you can keep it simple and just pile it up and leave it alone until it breaks down. There are people who put the stuff directly in their gardens and don't have problems. There are people who have damaged their garden by putting the stuff directly in it without composting it first. See some of the complicating factors I mentioned above.
If you want to study the issue and come up with a way to go directly on the garden, you can. If you want to put it directly on the garden without understanding the issues and trust to blind luck, you can. If you can find someone who is successfully putting it directly on the garden and mimic their procedure (probably not too bad a way to go whether they have developed a process through blind luck or actually understand the issues. I fit works for them , great!), you can. Or you can compost it first.
Now I'll get more directly to your question. One of the factors in how fast something composts is the nitrogen-carbon ratio. The poop provides nitrogen and the litter provides carbon. Naturally there is a perfect ratio of carbon to nitrogen for perfect composting but its very difficult to determine what your actual ratio is. Usually the litter and poop you take out is high in carbon and low in nitrogen. Usually. It depends on how often you clean it out, what materials you use, how many chickens you have pooping in what sized coop, whatever. It's different for all of us. In your case, it may actually not be that bad. Too much nitrogen is not that bad in a compost heap but too much carbon slows down the process. It will still compost but it will take longer. Adding more straw will add carbon, probably slowing down the process. Adding more straw or anything will not make the final product any easier to spread.
I don't have one of those bins. I just pile it in a pile and turn it two or three times before I use it in the garden. I have three piles, one composting, one gathering high nitrogen stuff like kitchen wastes and straight chicken poop, and one gathering high carbon stuff, like bean vines, corn stalks, and tomato plants. I don't sweat the nitrogen-carbon ratio other than trying to get a lot of nitrogen in there.
I'd imagine for the bins to work close to advertised, you would need to get the nitrogen-carbon ratio right, leave it alone until it heats up then cools back down, turn it at the right time, let it heat back up and cool down, and maybe go through this cycle again. The moisture content would have to be perfectly maintained, not too wet and not too dry.
I don't know which method would work best for you. Too many factors to consider that you know and we don't. I'd think a bin might work pretty good for you if your volume is low enough to fit. With you cleaning that area weekly, I'd think that if you go straight in a bin every week that the stuff in there would not have time to complete composting before you add more stuff so you'd never be able to empty it. Maybe store the fresh stuff in a big trash can or such until the batch in the bin is ready. If you get a bin with two sides and they are big enough, problem solved. I save the empty feed bags and store the completed compost in them until I need the compost.
I don't know if this helps or just makes it more complicated for you. Good luck!!!