Thanks, Scott. As you wrote this I was at the hardware store around the corner buying a 10" socket extender, a 13mm combo wrench, and 2 packets of JB Weld. One packet worth is currently setting in place on the tank seam, the other is in my tool bag for the road.
Based on where the leakage and visible damage was, I think what failed was the welded seam at the bottom of the tank (on the half that hangs to the left of the bar), not the welded seam between other parts. But I applied compound at a few other points that looked like they were degrading.
The tank is swiveled back (leaning against the pillion) but still connected.
I'm going to sit with it for an hour until it looks like the epoxy will hold its shape, then gently set the tank back in position and leave it to cure overnight. I'm hoping that by 7am (15 hours) it will be cured enough to reassemble and test.