Broke down and bought an antenna analyzer (MFJ-259B) and figured out what blew the tuner. The little 300W MFJ wattmeter, when connected in the line to the antenna, makes the SWR go through the roof. Remove the wattmeter from the equation and everything is fine.
I'd have been better off buying the analyzer in the first place rather than the wattmeter, I suspect. But live and learn.