OK, here are the promised pictures...
First, here's a link to the manufacturer's (Heng's industries) site, since I forgot to take a photo with the cap still on:
Attached below you'll find my pictures of my vent, with the cap off.
The first just gives context. This is the vent over the bathroom in the 19. I have already removed the top rain cap.
The second shows the ABS pipe inside this vent cover. It goes up behind the rounded corner of the shower stall, through the roof of the trailer. The Heng's cover provides a finished look to the black ABS and provides a rain cover.
But under windy conditions, rain can blow under the cap. The third the fourth photos look down inside the Heng's cover. There's space between the Heng's cover and the ABS pipe. Water can accumulate here.
(These also gives you an idea of how tight the space is if you're going to try to repair the caulking. You need to get down 3 or 4 inches to where the ABS comes through the trailer roof, but you've only got 1/2" or so between the ABS pipe and the rim of the Heng's vent cover.)
In the fourth picture you can see some of the caulking putty at the bottom of the ABS pipe. This is intended to keep water from entering the trailer if it gets between the ABS and the Heng's cover. But as I found, the putty had migrated up the pipe (presumably due to everything flexing with movement or temperature changes).
It no longer provides a seal, and any water that gets in this space runs down the ABS pipe, between the shower wall and the wall beside the dinette, into the guts of the trailer. I'm not sure where it goes, exactly, but the EMS and converter are located close to that pipe. And as the trailer changes position and particularly angle, the course of the water might change.
I'm not sure how it could have gotten over to the furnace (other side of the shower), but maybe the furnace problems I was seeing were caused by the electrical connections to the converter. The unusually high power drain did suggest a partial short, so maybe it was the furnace power circuit that was shorted, so that the furnace itself wasn't seeing sufficient voltage. (I'm not suggesting a full zero-resistance short (that would pop the fuse), just a bit of water letting a trickle of power through -- enough to drop the voltage at the furnace.)
Anyway, that's my theory.
Thanks for the suggestions for fixing the caulking in there.