OK good to know that you have the big predators. Scary as all get out!
So small budget lends to salvaged materials. My coop was built with reclaimed metal roofing and reclaimed shipping crates for the long sides. I am all for reclaimed items.
Take your time, get a drawing done on graph paper, make a plan.
Once you have a plan you can start gathering what you need. STICK to the plan. Deviations makes for difficult work a rounds later.
Here we have places that specialize in seconds, overbuys and reclaimed lumber. Look for those as sometimes great deals are there. I got siding for 7 bucks a sheet, normally 35 a sheet.
Wondering if a lean to style would fit your needs with the special needs bird.
This link will take you to a Wichita style coop. Not a fan of the size of this one but it has some nice features. I like a walk in coop for a ton of reasons.
OK so you are not a builder so take your time. Measure a bunch.
Will you have help in building a coop?
What are the dimensions of the current run you want the coop onside? I think you posted it but there are a lot of posts to search through to find it.