I will ignore the distinction here between the classifying space as a topological space or as a simplicial set. Fix for once and for all the adjunction $sSet \leftrightarrows Top$. Then giving the classifying topological space is 'just as good' as giving the classifying simplicial set. The 'classifying space' of a category qua simplicial set is then its nerve.
Consider the example of the category $Set$, which is a large category whatever foundational workaround you use. The nerve of $Set$ is not a functor $\Delta \to Set$, so one is forced (no pun intended) to use some sort of formalism that allows you to 'jump up a universe'. One could take NBG set theory, with classes and sets, and the nerve of $Set$ will be a simplicial class. Or one could take algebraic set theory (which is a categorical axiomatic formulation of sets and classes), or resort to Grothendieck universes or models arising from inaccessible cardinals, as Martin says in the comments.
I suppose one could try to work with simplicial classes informally, as a collection of sentences in first order logic, and the (explicitly given) maps between them, but I'm not sure how you can talk about class functions at this level. The only hope is that the face and degeneracy maps of the nerve of a category are very easy to describe, but I'm not an expert in that area.