Whoa... this seems to have done it. (I think.)
Thanks a lot for the tip! Funny, that's how I had it connected with the previous monitor and I didn't have trouble with the clamshell mode.
FWIW right-side power didn't help, nor did the turbo switcher AFAICT (I'm not doing anything CPU-intensive) -- but as soon as I put the monitor in the left and the power in the right the fans dropped way down. Still running, which is annoying, but not especially loud.
I find it maddening that a $3600 computer has a problem like this, and Apple doesn't even tell you about it.