Save and restore multiple layers within a Photoshop action that flattens

Sheepishly, I must admit that the solution was something I didn't think worked—actions will kind of reliably work making new documents, doing stuff in them, closing them, etc, as long as you have no other documents open. So I just do that now. Pasting the contents back into my original document often means things end up off-centre, but that's manually corrected and it's not too bad.