I need to transfer power wirelessly within a small SS tube that is not much larger than the diameters of the Tx and Rx coils. The coils are both inside the tube and are co-axial with it. Coils are 30mm diameter, although I'm also looking at 20mm coils with a max separation of 10mm ideally. Tube is about 40mm. Is this likely to cause a problem?
I had a similar problem which was solved empirically by trimming/shaving off parts of the SS that was not needed. The Rx winding was wound around a SS former but between winding and tube I laid ferrite rubber sheets (about 1mm thick). This largely reduced the problem but there were still some losses - you could feel the heat after a few minutes. This was resolved sufficiently by chamfering off extra bits of SS and it worked fine.
The transmit coil surrounded the RX coil so it's a little different to your set-up. What you might find is that an internal ferrite sheet will largely stop the flux causing significant eddy currents in the SS. I used a sheet from Ferroxcube but i don't think they make it any more but Wurth make one that is dead easy to cut and trim to size.
If the coils are more than (say) 5mm away from the SS, things do get easier. On another job it was end-to-end facing PCBs with spiral coils. The diameter was about 50mm and the SS was 5mm away and didn't pose a problem at all. In all cases I used 600 kHz.