The cause was probably a mod, I'll detail below:
It turns out that the problem was a custom animation I had in place for my unarmed power attacks. I lost track of which animation mod I was using, but I don't think it was too relevant for the purposes of the bug trigger. Basically, removing the specific animation for the power attack that wasn't working (in this case, hand to hand left), completely fixed the issue. After removing it, saving, and trying to add the same animation again, power attacks costed nothing again. However, after adding a custom animation from a different unarmed combat mod, it turns out that one works. Huh.
Anyway, I still have no idea what actually caused it to occur in the first place, but probably this would work for anybody who is encountering this issue with their power attacks. If you don't know exactly what the cause is, try removing any corresponding custom animations first.
It is notable that I am using FNIS for PC-exclusive animation paths, and the reason it worked on a new character is because my test race was default Redguards. This is what tipped me off; making a new character with a custom race had the same animations, and thus the same problem.
EDIT: I had to do a little digging, but I confirmed that the problem animation was from the http://www.nexusmods.com/skyrim/mods/9421/? mod - specifically, it was the Round Kick, which is the right-hand power attack by default. I set mine to left since I was using animations from that and Unarmed Warfare. Presumably, using this with FNIS caused problems because of something specific to that animation; the author of FNIS speculated that it may have been a bit shorter than the default or something.
Curiously, though, I could not replicate the problem again after switching my animations around initially. Don't know why. Skyrim is weird.