Suppose I route coax through a strong DC magnetic
field, such as in an MRI, with the magnetic field
perpendicular to the cable axis. Will the
current distribution in the center conductor then
become biased "up" with respect to the
field (right hand rule), and the current distribution
in the shield then become biased "down,"

I think you're distorting the Hall Effect here. One side of each
conductor will have a positive charge and the other side a negative
charge, and there will be a potential across the shield and across the
inner conductor. But at the ends of the cable these effects are gone.

thereby destroying symmetry?
If so, will the coax then radiate and become lossy?

No. Superposition of fields. Even for an AC magnetic field.

If you have an alternating AC magnetic field you can induce a current
in both the shield and inner conductor, but these will match. If
you've got ground loops you may misinterpret this as a signal. Don't
have ground loops.

Tim.