If you consider those predator-proof enough so you can always leave the pop door open, they will always have use of the entire enclosed area when they are awake. And if you let them range outside every day instead of leaving them locked in that much of the time when they are awake you should have enough space. That may work fine most of the year but winter in Missouri may test that plan.
The way I understand it you are not planning you are not expecting them to sleep on that outside roost in the run. You may find that they don;t always do what you plan. Good luck with that.
To your question about how to raise the roosts inside the enclosed coop section. I think I'd want them to run the same way they do now to minimize the chance of them pooping in the nests. Those nest covers may or may not be necessary. On the wall side just do what was done, Screw in a support to the wall to support that end of the roost. On the nest side drop a support from above. I don't know what your framing in the roof looks like, can you drop something from that framing? Or put in a high horizontal that you can drop a vertical from.
You could build a frame in there and support the raised roosts from below, but I think in that tight space you will have enough trouble with access for cleaning. That would make it worse.
I personally think you are making too much daily work for yourself but that's for you to work through. Good luck.