I'm not sure I can exactly answer your question, but I can share my experience because I did exactly the same thing. I built a 3'x12' winter coop across the back of the "extra bay" of my barn - the 12x12 bay also has my workbench, storage shelves, and hay/bedding storage bins. The six chickens have been there just about 2 months, since mid December, so that's all the track record I have.
My barn is a center-aisle barn, 36' wide x 24' deep, 10' high first level with a hay loft overhead so a fair amount of enclosed air.
I built my coop with a solid ceiling (used for storage overhead), at 6' tall for most of the width but just 4' tall on one end (about 3' wide) (because I had some larger tools I wanted to store there; you see the partition between that shelf and the tall part of the coop at the top left of the picture). Nest boxes and future pop door are in the short section; two 3' roosts are to the right. There's a rough-sawn 1x12 dropped into slots across the doorway to hold the bedding in, removable for cleaning.
My ventilation is the 6'H x ~5.5'W doorway, plus a ~3'W x 8"H strip across the top of the wall to the right of the picture. Bedding is shredded paper/cardboard. Currently the chickens are let out the main door in the morning to run across the aisle and spend their day in a stall (also shredded paper/cardboard bedding). Or most of the day - they hop out of the stall to return to the coop to lay (as they don't seem to like the makeshift nest box I gave them there), then return to the stall, and they will also hop out to dig around in the half of the aisle I haven't floored yet. Construction of a covered outdoor run is planned for next summer.
My thoughts:
The air in my coop seems no less fresh than the air throughout the barn, so that doorway is giving plenty of ventilation. Granted, so far, most of the time it's been in use the temperature has been well below freezing, but we've had a few thaws and I haven't noticed an issue.
If you can use the storage, a solid ceiling is great, and would probably be easier to build than a hardware cloth ceiling (but that might just be me, because I hate trying to get the stuff looking smooth). But hardware cloth would be fine if you don't want a surface up there collecting dust.
^^ Because with the top 3' of your walls hardware cloth, I can't imagine it will make a difference to the dust level whether the top is solid or mesh. With my one side ventilation (no cross-ventilation) I've got plenty of dust; it looks like you will have three (or is it two?) sides so the air will move through the coop & carry the dust with it. (Of course my birds are also in the same building 24/7 so only maybe 1/2 the dust is from the coop portion.)
Our llamas spent the nights in the stall last winter (pre-chicken) and that created a lot of dust, too, so I'm used to it. I do have to blow off or dust off things that have been sitting a while! (The llamas have their own run-in shed now.)
I do really like having the coop in the barn. I'm completely fine with the dust. It's just so nice to have most of the work under one roof - get the llama feed & hay, get the horse feed & hay, move the chickens & check their feed, morning and night, all in from the weather.
Also, the air is fresh without having perceptibly moving air. When I go in the barn on the really cold mornings that dead air (in a good way - fresh but still) is so much milder-feeling than the bite of the outside air. I'm not sure it's measurably much warmer, but it feels it.
This is the only picture I have on this device. Hope something in this post helps!
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