The stage $L_{w+w+1}$ 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 $L_{w+w+1}$, Sets are elements of $L_{w+w}$ and proper classes are elements of $L_{w+w+1}$ that are not elements of $L_{w+w}$. $L_{w+w}$  interprets $V$. Clearly it is provable *within* $L_{w+w+1}$ that for each $n=0,1,2,..,w$ we have $L_{w+n} < L_{w+n+1}$ where < is strict subnumerousity, the largest cardinality "as seen inside $L_{w+w+1}$" is the cardinality of $L_{w+w}$, we can define cardinality within $L_{w+w+1}$ as:

 $|x|=y \iff \exists F (F:y\to x \wedge \text{y is a bijection})\wedge (\text {y is a finite von Neumann} \lor \exists i (\text {x is infinite} \wedge y=L_i)$.

In English the cardinality of a set x is the finite von Neumann ordinal that is equinumerous with x or the infinite stage of the $L$ that is equinumerous with x. So we define $\aleph_n$ as $L_{w+n}$ for $n=0,1,2,3,...,w$. 
 
Now because $L$ proves the Generalized Continuum Hypothesis so it is provable within $L$ that all cardinals strictly smaller than $\aleph_w$ are either $n$ or $\aleph_n$ for some natural $n$ and each one of those is the cardinality of some element of $L_{w+w}$ and each one of those is itself an element of $L_{w+w}$. The set of all stages in $L_{w+w+1}$ is a proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is provable within $L_{w+w+1}$ to be strictly smaller than $\aleph_w$ would be provable within $L_{w+w+1}$ to be of cardinality $n$ or of cardinality $\aleph_n$ for a natural n, and 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.