I reset the watch, and paired it as new (not restoring from backup).
It did not help; the stopwatch (when running) always appears when I lift my wrist, regardless of which watch face I have shown, what complications it has, or how long my wrist has been down (far past the 'within 2 minutes of last use' setting I have enabled for 'last app').
An aside note: resetting the watch as new did fix a few other problems I had, such as phantom alarms going off no matter how many times I cleared them, or that my activity coaching alerts never came up.
So, resetting the watch as new does fix some oddities, but this particular issue appears to be deliberate or a bug that's not fixable even with a clean reset.