re:
> Would be very grateful for some pointers on how I can make this work

Is there any way for you to make it into a solid rather than starting with an open surface like that? That's one thing that may help.

The other thing you could do to make progress would be to use Edit > Separate on it to break it apart into individula surfaces, then use Offset or shell on those individual pieces which will succeed and then work on extending planes so they intersected each other, use Construct > Curve > Isect to intersect those planes to get extended boundaries and then build new planes from those boundaries.

Or you might look at some other CAD programs like maybe ViaCAD Pro might potentially have a better surface thickening function than MoI currently has, if you import your model into there that might work.

- Michael