The stage $L_{2w +1}$ of the constructible universe of ZFC would serve as a model of $MK^-$ as far as the first two conditions are concerned, but it would fail the third condition. Classes would be interpreted as elements of $L_{2w +1}$, Sets are elements of $L_{2w}$ and proper classes are elements of $L_{2w +1}$ that are not elements of $L_{2w}$ . $L_{2w}$  interprets $V$. The largest cardinality of an element in $L_{2w+1}$ is ${\aleph_w}$, all strictly smaller cardinals are either $n$ or $\aleph_n$ for every natural $n$ and each one of those is the cardinality of some element of $L_{2w}$. The set of all stages $L_{i}$ for i<$2w$ is proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is strictly smaller than $\aleph_w$ would be of cardinality n or of cardinality $\aleph_n$ for a natural n, so equinumerous to a set. However the third condition cannot be met since the set of all singletons of elements of any proper class would be a proper class too. And so this remains a partial answer.