Update: I ended up splitting the outer parts from the inner parts, which still couldn't pass the Shapeways requirements, but i.materialise accepted them both (so far), so hopefully I'll see those in mid-May. I'm adding those two models to the bottom of this page.
I really like how this is evolving, but I still haven't quite managed to make it printable. I'm thinking my next step might be to make a tool that will enlarge the middle small parts and shrink the outer large parts. [Turns out proportional editing can work as such a tool.]
Funny story, all these years I've been playing with spherical inversions I've been using a very simple specific case of spherical inversion, but last night I implemented the whole thing, which allows me to change the radius of the inverting sphere and origin of it.
The two images below are from Shapeways' tool to illustrate the parts that are too thin to be printed in stainless steel. So my next step will be to create a tool that allows me to scale the object varying by radius, so I can make the small parts bigger without making the big parts too big. And I suppose that could be a handy tool elsewhere too!