After long last I finally migrated my Dockered Unifi Network instance into their Unifi OS. I naturally put that in a Docker as well but I've been putting off this move as Unifi OS has a buttload of other junk too that I have really not wanted to deal with.
Basically I just wanted to have the system the way it was, with just the Network "app" in it. By not logging into an Unifi account during the setup, it landed me into the dashboard with an extremely barebones system with just the Network installed. I restored the old stand alone Network app's backup there and looks like it's exactly the way it was, except I now have an upgrade path when they discontinue the standalone app in the future.
I used this image, it was trivial to set up for someone who had already dealt with the one that just ran Network:
