I've isolated it to the bottom bracket:
- The noise also happens when I'm standing, so it's not a saddle creak.
- The noise is present if I spin with only one foot, either side; so it's not a pedal problem.
- I checked the crank bolts and the arms are not loose.
- I checked the chainring bolts and they are not loose.

It happens at the same spot in the pedal stroke, forward or backward. It's a click/tick as the crank moves past a certain position. It doesn't happen without pressure on the pedals; just spinning them in the stand won't do it, but if I apply the rear brake and use some force on the cranks I can make it happen.
The BB doesn't feel as though it's sticking at that spot when I spin it by hand. It's a less than 1yr old UN54, so I don't think it's a problem with the bearings.

I'm going to take it out and regrease the threads and see if that helps.

The Question: Would riding all fall/winter/spring through the soaking rain eventually deteriorate or wash away the grease I used when origainally installing the part, leading to this noise?

Yep. Most likely, a thread is clicking somewhere. I see you checked the pedals. It's best if you can swap the pedals out entirely to test. Certainly try regreasing each pedal thread, even if it doesn't seem like that's the issue.

The next step would be to regrease all threads, doing one set at a time for testing purposes, in case it happens again.

What sort of crank and bottom bracket are you using?

EDIT: I see it's a UN54. It's unlikely the bottom bracket itself is clicking, but the threads may very well be.

It's a Sugino XD700 crank.

While I'm greasing things up, it can't hurt to have the pedals off and re-grease them, too. Once I ruled them out as the culprit, I shouldn't have been so quick to write them off. Some preventive maintenance will keep them from becoming the next noise I have to track down.

Torque the BB to specs.