My last click that I was certain was either BB or, possibly saddle-rails / seatpost turned out to be…
…headset bearings! Or, more specifically, them moving fractionally on the seats. I packed the whole lot in with the thickest grease in the garage and tightened it all up as firmly as it would take and it went away.
This always seems to be worse on plastic bikes – any noise seems to echo around the frame and be nearly impossible to track down. You’d have thought that a bike would be a trivial thing to find a noise on, but I’ve been stumped a time or two. The one before that was rattly valve stems, which I was sure was a rogue loose spoke (at least that one sounded like it was coming from the wheel!)