OK - I'm a little disappointed to say I don't have a culprit to report.
I reinstalled vanilla Skyrim, then turned off all my mods. I reinstalled and added each one at a time, and FNIS worked like a charm. Probably the only change I made was to accept Immersive Animations rather than the XPMSE animations, per the STEP guide on Dual Sheathing Redux.
So the good news is it is working! The bad news is, I'm not entirely sure why...
