Can there be a model of "$MK-Limitation$ $of$ $size+Subsets-Union$" having a proper class $P$ that is strictly smaller than the class $V$ of all sets and yet $P$ being equinumerous to some set?

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