Dear Ben,
I have no experience with the multi-block inspector feature and wouldn't know where this feature needs to be put for the VTK-format.
But what you can do is, export a field that assumes different values on different objects, e.g. with DomainConstantCoefficientFunction, i.e. CoefficientFunction([1,2,3,4,5]) . Afterwards you can use ParaViews filters to select only certain objects...
Best,
Christoph