If you set the window’s collectionBehavior to:

    NSWindowCollectionBehaviorStationary | NSWindowCollectionBehaviorCanJoinAllSpaces

…that should give it roughly the same behavior as the system desktop window. If you want to have things like different windows for each space, that might require the brittle hacks you’re talking about :-).

By the way, I meant to say that your approach is super cool too, and the blur and shadow effects look great.