PROBLEM SOLVED
We bought a new Wiimote today and tried again. Everything worked perfectly!
So just to sum up, if the IR Camera View is erratic, you get a 56% battery reading (with batteries that are not actually at 56% power, of course), only IR lights 2 and 3 are usually active, then you have a faulty wiimote.
By the way, I didn't get any useful information out of Nintendo. I guess they haven't really heard of fake wiimotes. No matter. Just be sure you buy your wiimotes new from a reputable retailer, NOT from an online seller on Amazon or Ebay or et cetera.
Hope this is helpful to anyone who has this problem in the future.
