Thanks - I think I have it! (At least as far as Zip disks go...)
If I have Iomega tools running, once I insert the Zip disk into the drive it sees the disk and tags it as an unknown format. For whatever reason this in turn stops OS 9 from showing the unrecognised disk format screen, and I can open MESA and see the contents of the disk from the MESA "Disk" window.
A bit clunky and the sequencing of turning everything on and inserting disks is important, but once it's up and running MESA works like a charm!
Now I'm off to lie down...