Ok, I am an idiot. Forest for the trees and all that.
Drag control from left panel to right panel. Switch left panel, drag from right panel to left.
The key is that you can only drag from one visible panel to another visible panel, as was told to me earlier, but for some reason didn't click.
It would be nice if in a future update, if you drag a control to a page tab, it switches to that page and then you can drop it there, kinda like when you're organizing icons on an iOS device as you drag to the edge, it switches pages.
Anyway, all sorted now, thank you everyone for your patience!