The Kelley-Morse set theory can be thought as the "full-secondorderification of $\sf ZFC$", where we switch from sets to classes and allow the comprehension schema to include quantifiers on class variables.

As in the usual case of class-set theory, sets are exactly those classes which are elements of other classes. So we can consider this as a one-sort theory and have a definable predicate $\mathrm{Set}(x)\iff\exists y(x\in y)$.

The "standard" models of Kelley-Morse set theory are $V_{\kappa+1}$ where $\kappa$ is a strongly inaccessible cardinal, and then $\mathrm{Set}$ is interpreted exactly as $V_\kappa$.

But we can use the Lowenheim-Skolem theorem to obtain a countable model of Kelley-Morse. The trick here is that we keep "enough" classes to satisfy the impredicative comprehension schema, but the model is still countable.

My question is as follows:

Let $M$ be a model of Kelley-Morse What sort of limitations do we have on the cardinality of $M$ and $\mathrm{Set}^M$? Specifically, can we have a model with countably many sets and uncountably many classes? Does the answer change if we assume that $\mathrm{Set}^M$ is transitive?