Lo, the pump was flagged for extra screening and, shortly thereafter, so were the bottles. The TSA agents were cheerful and friendly as they did it, rather than grumpy and surly. The process was quick, just some kind of sensor wand passed over the outside of the pump and each bottle. Results were negative and I was on my way.
So... it was worrying that I couldn't get a straight or consistent answer to my question, but I gotta give the screeners props. They're not all as bad as the horror stories1 make them out to be.
1- And, to some extent, personal experience. I've never had a Horror Story, but I've been barked at and scowled at plenty. Maybe being aggressive is seen by their supervisors as Taking Security Seriously? I don't know.