Giving Wisdom didn't work, as I thought. The Wisdom stat is completely gone, it does not even have a UnitStat definition anymore.
It isn't something unique with the Imbue Champion spell either, same problems occur if it is cast on a non-Champion unit (after being modified to allow it, of course). Furthermore, giving every Champion starting Essence (thank god for find/replace...) lets them cast spells right off the bat, no problems. A carbon-copy of the Imbue Champion spell also works properly when cast on Champions, but not on other units.
My guess? A unit needs to be a Champion before it will even check to see if the unit has Essence in Tactical Battles. Or a bug. Oh well...
Hmm... thinking on it, I don't suppose anyone would know what exactly causes a Champion to be considered a "Champion", would they? Maybe that would fix the problem, although I suppose that has a chance to create other problems...