On the apron, I like taking the turf off, laying the wire down and attaching it, then replacing the turf. I did not do that once for a dog pen and the wire got into my weed eater. That’s still a lot easier than digging down a foot and, in my opinion, more effective.
I’m not clear on how they are going to be secure in the coop without locking them in unless your run outside is really predator proof. It can get expensive to build a run like that, especially a decent sized run. I have a short elevated tunnel, maybe 2’ long, between my grow-out coop and the run. I used a guillotine type door so I could block it off. My run is pretty secure but this makes the coop really secure.
One flaw in my design is that I don’t have a good way to put chickens in that grow-out coop from the run and lock them in the coop without walking around to get outside the fence. I keep meaning to change that but haven’t gotten around to it. You might want to give some thought on how you would do that with your set-up. Hopefully your walk will be less than my 200’. I really need to get around to fixing that.
Your coop needs to perform a few functions, mainly protection from weather and predators. Different barns provide different levels of protection for both. Up where you are barns are usually pretty weather tight, unlike some further south, so all you should need is a wire coop to keep them contained. It’s really hard to make a big structure like a barn truly predator proof. They are just too big and too many opportunities for a weakness though yours may be really good. It’s hard to say from here.
If I were building your coop I’d frame it up and cover it with 2”x4” welded wire. I’d not use netting since a raccoon or even a bobcat can get through that if they want. I’d take the welded wire to a solid surface, not leave any openings for a climbing or flying predator like an owl. Rats, snakes, and many members of the weasel family can get through that, but to stop them you pretty much need to go with ½” or ¼” hardware cloth. That can get really expensive. I’d be satisfied with the protection 2x4 wire gives me with one exception.
Raccoons have been known to set just outside a fence and pull a chicken through the wire piece by piece. At night when they are roosting they are really vulnerable. If your roosts butt up against the welded wire, I’d suggest using chicken wire or hardware cloth to protect an area where a raccoon can’t reach through.
There is another potential problem with 2x4 wire if you raise baby chicks in there. They can walk right through it. If you plan to raise young chicks in there put a finer mesh inexpensive wire like chicken wire around the bottom foot to 18”. That will help keep raccoons from reaching through too. Or if you want to make your entire coop more secure, line the inside wire with chicken wire. It’s not that expensive and does improve security. Not all chicken wire or other wire mesh is created equal. The openings can be different sizes and they come in different gauges. Pick something in your budget that does what you want.
When I made my elevated tunnel, I used a piece of plywood a foot wide for the floor and bent ½” hardware cloth to make a tunnel about 12” high. Full-grown full-sized roosters can and do use that but if I were doing it again I’d make it at least 18” high so they don’t have to scrunch down and walk kind of funny.
In your climate a wire coop in a barn sounds great. It doesn’t have to cost much, you should be able to make it pretty large, it should have great protection from the elements, and will have great ventilation. Sounds perfect to me.