I dug up a real mumetal shield, attached extension leads to the filament transformer and experimented.
I can now 100% confirm the transformer does radiate enough to effect the image.
Rotating 90 degrees helps quite a bit but does not eliminate it. Unfortunately, there's not enough room to mount it this way.
Putting the shield over it helps even more.
Running the tuner off pure DC is about the same as using the shield. There is still some 60Hz getting into the horizontal sync but it's not very noticeable.
I'll see if I can mount the transformer under the chassis.