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The stage $(V_{w+w+1})^L$ 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 $(V_{w+w+1})^L$, Sets are elements of $(V_{w+w})^L$ and proper classes are elements of $(V_{w+w+1})^L$ that are not elements of $(V_{w+w})^L$. $(V_{w+w})^L$ interprets $V$. Clearly it is provable within $(V_{w+w+1})^L$ that for each $n=0,1,2,..,w$ we have $(V_{w+n})^L < (V_{w+n+1})^L$ where < is strict subnumerousity, since each $(V_{i+1})^L$ is the constructible power set of $(V_i)^L$ and $L$ is a model of $ZFC$, now the largest cardinality "as seen inside $(V_{w+w+1})^L$" is the cardinality of $(V_{w+w})^L$, and we can define cardinality within $(V_{w+w+1})^L$ as:

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

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 $(V_{w+w+1})^L$ that is equinumerous with $x$. So we define $\aleph_n$ as $(V_{w+n})^L$ for each $n=0,1,2,3,...,w$.

Now because $L$ satisfies 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 $(V_{w+w})^L$ and each one of those is itself an element of $(V_{w+w})^L$. The set of all $V_i^L$ stages in $(V_{w+w})^L$ is a proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is provable within $(V_{w+w+1})^L$ to be strictly smaller than $\aleph_w$ would be provable within $(V_{w+w+1})^L$ to be of cardinality $n$ or of cardinality $\aleph_n$ for a natural $n$, and so equinumerous to a set. So $(V_{w+w+1})^L$ satisfies $MK^-$ plus the first two conditions. 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.

The same argument would hold in any model of $ZF+GCH$, the stage $V_{w+w+1}$ of the cumulative hierarchy of that model would be a model that satisfies all axioms of $MK^-$ plus the first two conditions of the question. However it still fails the third condition. So it is also a partial answer.

The stage $(V_{w+w+1})^L$ 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 $(V_{w+w+1})^L$, Sets are elements of $(V_{w+w})^L$ and proper classes are elements of $(V_{w+w+1})^L$ that are not elements of $(V_{w+w})^L$. $(V_{w+w})^L$ interprets $V$. Clearly it is provable within $(V_{w+w+1})^L$ that for each $n=0,1,2,..,w$ we have $(V_{w+n})^L < (V_{w+n+1})^L$ where < is strict subnumerousity, since each $(V_{i+1})^L$ is the constructible power set of $(V_i)^L$ and $L$ is a model of $ZFC$, now the largest cardinality "as seen inside $(V_{w+w+1})^L$" is the cardinality of $(V_{w+w})^L$, and because $L$ satisfies the general continuum hypothesis then we can define cardinality within $(V_{w+w+1})^L$ as:

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

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 $(V_{w+w+1})^L$ that is equinumerous with $x$. So we define $\aleph_n$ as $(V_{w+n})^L$ for each $n=0,1,2,3,...,w$.

Now because $L$ satisfies 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 $(V_{w+w})^L$ and each one of those is itself an element of $(V_{w+w})^L$. The set of all $V_i^L$ stages in $(V_{w+w})^L$ is a proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is provable within $(V_{w+w+1})^L$ to be strictly smaller than $\aleph_w$ would be provable within $(V_{w+w+1})^L$ to be of cardinality $n$ or of cardinality $\aleph_n$ for a natural $n$, and so equinumerous to a set. So $(V_{w+w+1})^L$ satisfies $MK^-$ plus the first two conditions. 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.

The same argument would hold in any model of $ZF+GCH$, the stage $V_{w+w+1}$ of the cumulative hierarchy of that model would be a model that satisfies all axioms of $MK^-$ plus the first two conditions of the question. However it still fails the third condition. So it is also a partial answer.

The stage $(V_{w+w+1})^L$ 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 $(V_{w+w+1})^L$, Sets are elements of $(V_{w+w})^L$ and proper classes are elements of $(V_{w+w+1})^L$ that are not elements of $(V_{w+w})^L$. $(V_{w+w})^L$ interprets $V$. Clearly it is provable within $(V_{w+w+1})^L$ that for each $n=0,1,2,..,w$ we have $(V_{w+n})^L < (V_{w+n+1})^L$ where < is strict subnumerousity, since each $(V_{i+1})^L$ is the constructible power set of $(V_i)^L$ and $L$ is a model of $ZFC$, now the largest cardinality "as seen inside $(V_{w+w+1})^L$" is the cardinality of $(V_{w+w})^L$, and we can define cardinality within $(V_{w+w+1})^L$ as:

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

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 $(V_{w+w+1})^L$ that is equinumerous with $x$. So we define $\aleph_n$ as $(V_{w+n})^L$ for each $n=0,1,2,3,...,w$.

Now because $L$ satisfies 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 $(V_{w+w})^L$ and each one of those is itself an element of $(V_{w+w})^L$. The set of all $V_i^L$ stages in $(V_{w+w+1})^L$ is a proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is provable within $(V_{w+w+1})^L$ to be strictly smaller than $\aleph_w$ would be provable within $(V_{w+w+1})^L$ to be of cardinality $n$ or of cardinality $\aleph_n$ for a natural $n$, and so equinumerous to a set. So $(V_{w+w+1})^L$ satisfies $MK^-$ plus the first two conditions. 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.

The same argument would hold in any model of $ZF+GCH$, the stage $V_{w+w+1}$ of the cumulative hierarchy of that model would be a model that satisfies all axioms of $MK^-$ plus the first two conditions of the question. However it still fails the third condition. So it is also a partial answer.

The stage $(V_{w+w+1})^L$ 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 $(V_{w+w+1})^L$, Sets are elements of $(V_{w+w})^L$ and proper classes are elements of $(V_{w+w+1})^L$ that are not elements of $(V_{w+w})^L$. $(V_{w+w})^L$ interprets $V$. Clearly it is provable within $(V_{w+w+1})^L$ that for each $n=0,1,2,..,w$ we have $(V_{w+n})^L < (V_{w+n+1})^L$ where < is strict subnumerousity, since each $(V_{i+1})^L$ is the constructible power set of $(V_i)^L$ and $L$ is a model of $ZFC$, now the largest cardinality "as seen inside $(V_{w+w+1})^L$" is the cardinality of $(V_{w+w})^L$, and we can define cardinality within $(V_{w+w+1})^L$ as:

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

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 $(V_{w+w+1})^L$ that is equinumerous with $x$. So we define $\aleph_n$ as $(V_{w+n})^L$ for each $n=0,1,2,3,...,w$.

Now because $L$ satisfies 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 $(V_{w+w})^L$ and each one of those is itself an element of $(V_{w+w})^L$. The set of all $V_i^L$ stages in $(V_{w+w})^L$ is a proper class of cardinality $\aleph_0$ which is strictly smaller than $\aleph_w$, and every proper class that is provable within $(V_{w+w+1})^L$ to be strictly smaller than $\aleph_w$ would be provable within $(V_{w+w+1})^L$ to be of cardinality $n$ or of cardinality $\aleph_n$ for a natural $n$, and so equinumerous to a set. So $(V_{w+w+1})^L$ satisfies $MK^-$ plus the first two conditions. 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.

The same argument would hold in any model of $ZF+GCH$, the stage $V_{w+w+1}$ of the cumulative hierarchy of that model would be a model that satisfies all axioms of $MK^-$ plus the first two conditions of the question. However it still fails the third condition. So it is also a partial answer.