Thanks Eric -- you were right.
When both taps are on full, no squeal. When the cold is backed off to make the temperature right, full squeal.
One of the things I'll be doing is replacing the pipes to this 2nd floor bathroom. The guy who plumbed it in 1994 made a lot of really nasty looking solder joints which (based on the excessive pipe noise at certain points) are likely clogged with solder.
Upsizing the pipes and getting the clogs and superfluous bends out should really help boost the flow.
I'll let you know how it goes.