Can there be a model of the theory "$\mathsf{MK}-\text{Limitation of size}+\text{Subsets}-\text{Union}$" having every proper class strictly smaller than the class $V$ of all sets being equinumerous to a set (provided of course that the model has at least one proper class that is strictly subnumerous to $V$)?

Where $\mathsf{MK}$ is Morse-Kelley set theory, and $\text{Subsets}$ is the axiom asserting that every subclass of a set is a set.