The reason that meta-language solutions to the semantic paradoxes can't work is because to characterise the concept which generates the paradox (e.g. 'truth'), a language which is not in the hierarchy is required.

I would suppose that the theory of types is similarly vulnerable to a diagonalisation paradox based on the intensional definition of a set which depended on a feature of the hierarchy in such a way that the set became both typeless and typed?

I'm sure this must have been done, and I'll have a look.