The stage $L_{(\aleph_w+1)}$ 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 $L_{(\aleph_w+1)}$, Sets are elements of $L_{\aleph_w}$ and proper classes are elements of $L_{(\aleph_w+1)}$ that are not elements of $L_{\aleph_w}$ . $L_{\aleph_w}$  interprets $V$. The largest cardinality is ${\aleph_w}$, all strictly smaller cardinals are are either $n$ or $\aleph_n$ for every natural $n$ and those are already elements of $L_{\aleph_w}$. The set of all those cardinals is a 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.