Well... I'm $288 poorer; but the mobile bank vault is back on the road again!
Technician promptly identified the starter as the source of my problems. He noted that the device was drawing way too much current (though he did express some surprise afterward that the solenoid was in such good shape). Anyway, he recommended replacing the starter, which I promptly okayed. (It was the original starter--nearly 15 years old.)
1 hour of labor later--which included 25 successive and successful restarts of the engine--and my tech called with the "all clear".
Thanks to those who responded. Your combined expertise bolsters my confidence that this problem is now history.