Mu-metal probably won't help.

My understanding is that it should be high-nickel content metal, such as is
mu, and that without that content magnetic fields aren't much effected by
metals.

A copper shorting strap around the
transformer would. You can make that out of sheet copper, or even bare
wire and a lot of solder.

Or does solder serve as the poor-man's nickel?

Sounds like DIY mu-copper :-) You say "strap" which I interpret as not
full-height of the transformer. Should it be less-than the height of the
lams? Or is taller better?

Isn't that a lot of wire (a tertiary winding)?

Wrap the sheet around and bolt it under the feet of the xfmr? (Only need to
shield in 1 direction, really: forward.)

Or, um, move it farther away.

Yeah, thanks for that, John... :-)
--
Sparky