I found that I could move the wikipage controls easily, it was just in a separate menu from the one that moved modules.
Probably some of my other temporary failures come from not knowing where things are in the complex choice tree.
It probably wouldn't be worth putting the wikipage controls into a separate module, just to be more compatible with something else. Once somebody knows where the separate controls are, they know and aren't bothered by it again.
So now my original goal is complete except that I lack an easy way to automatically put the backlinks at the bottom of the wikipage. Someone suggested there should be a way to automatically run a contents template a long time ago, but I haven't seen any documentation that says it's been done.