Anybody have a suggestion for a starting point on this one, I'm totally clueless.

I will search my library to give you some hint. I'm sure I will find some good information on it, but that will take till tomorrow

I do not believe there will be any losses due to eddy currents.

A highly conductive material will sustain strong eddy currents for a long time, whereas the eddy currents in poor electrical conductors are weaker and decay rapidly, if any exist at all. The power from eddy currents increases as the square of resistivity divided by the permeability.

In fact, one way that motor manufactures increase efficiencies (by decreasing eddy current losses), is to increase resistivity by utilizing high(er) silicone content in the magnetizing steel.

To motorman:

There will be eddy currents in the stainless steel if the current is an AC current. There are people using induction furnances to melt stainless steel.

To gordonl:

unfortunately I haven't found  a book or an paper describing an simple solution for your problem, but let me describe the way I would approach the problem without access to FEM-software to solve the problem.

Calculate the magnetic field H at the position of the stainless steel part (H= I/(2*pi*r)) might be an acceptable approximation even if your conductor is not circular.

Calculate the voltage induced in  a loop on the outer surfcace of the stainless steel part, assuming that H is constant and that it is not changed by eddy currents induced.

Calculate the current density caused by the induced voltage.

Calculate the specific losses per volume.

This will give you only a very rough estimation but then you are able to decide if further calculations eg. FEM-analysis is necessary or not.