I claim that this is equiconsistent with "$ORD$ is Mahlo" (This is not the same as $Ord$ being actually Mahlo, as I have explained elsewhere). It turns out "$ORD$ is Mahlo" is a natural limit point for these kind of Ackermann/$KM$ based theories. Let $T$ be your theory.

First, the easy part. The consistency strength of "$ORD$ is Mahlo" $\ge$ the consistency strength of $T$. Let $C=\{\alpha|V_\alpha\prec W\}$, where $W=\{x|x=x\}$. Note that if $\alpha\in C$ and $\phi(\alpha)$, then $\exists\beta(\beta\gt\alpha\land\phi(\beta))$. The reason for this is that else $\psi(\alpha)\leftrightarrow\beta\text{ is the largest } \beta\text{ such that }\phi(\beta)$ would be a definition of $\alpha$, and so $V_\alpha\vDash\exists x(\phi(x))$, which is a contradiction. Now, if $\phi(\alpha)$ with $\alpha\in C$, then $\phi(\alpha\land\alpha\text{ is a cardinal})$ and so $V_\alpha\vDash\forall\eta(\exists\beta\gt\eta(\phi(\beta\land\beta\text{ is a cardinal}))$. Therefore, whenever $\kappa$ is inaccessible reflecting $V_\kappa\vDash T$, and the existsence of an inaccessible reflecting cardinal is equiconsistent with "$ORD$ is Mahlo."

Second, the hard part. The consistency strength of $T\ge$ the consistency strength of "$ORD$ is Mahlo." Let $C=\{\alpha|\phi(\alpha,p)\}$, and let $C^V=\{\alpha|\phi^V(\alpha,p)\}\cap Ord$ be club in $Ord$. Then you can see $\{\alpha|\phi^V(\alpha,p)\}$ is club (In the real class of ordinals); in particular $\phi^V(Ord,p)\land Ord\text{ is regular}$. Then we can find a non-empty class of regular $\kappa\in C^V$, and so $\phi^V$ when $ZFC+ORD\text{ is Mahlo}\vdash\phi\rightarrow\phi^V$. Therefore, if $M\vDash T$ then $V^M\vDash ZFC+ORD\text{ is Mahlo}$

I claim that this is equiconsistent with "$ORD$ is Mahlo" (This is not the same as $Ord$ being actually Mahlo, as I have explained elsewhere). It turns out "$ORD$ is Mahlo" is a natural limit point for these kind of Ackermann/$KM$ based theories. Let $T$ be your theory.

First, the easy part. The consistency strength of "$ORD$ is Mahlo" $\ge$ the consistency strength of $T$. Let $C=\{\alpha|V_\alpha\prec W\}$, where $W=\{x|x=x\}$. Note that if $\alpha\in C$ and $\phi(\alpha)$, then $\exists\beta(\beta\gt\alpha\land\phi(\beta))$. The reason for this is that else $\psi(\alpha)\leftrightarrow\beta\text{ is the largest } \beta\text{ such that }\phi(\beta)$ would be a definition of $\alpha$, and so $V_\alpha\vDash\exists x(\psi(x))$, which is a contradiction. Now, if $\phi(\alpha)$ with $\alpha\in C$, then $\phi(\alpha)\land\alpha\text{ is a cardinal}$ and so $V_\alpha\vDash\forall\eta(\exists\beta\gt\eta(\phi(\beta\land\beta\text{ is a cardinal}))$. Therefore, whenever $\kappa$ is inaccessible reflecting $V_\kappa\vDash T$, and the existsence of an inaccessible reflecting cardinal is equiconsistent with "$ORD$ is Mahlo."

Second, the hard part. The consistency strength of $T\ge$ the consistency strength of "$ORD$ is Mahlo." Let $C=\{\alpha|\phi(\alpha,p)\}$, and let $C^V=\{\alpha|\phi^V(\alpha,p)\}\cap Ord$ be club in $Ord$. Then you can see $\{\alpha|\phi^V(\alpha,p)\}$ is club (In the real class of ordinals); in particular $\phi^V(Ord,p)\land Ord\text{ is regular}$. Then we can find a non-empty class of regular $\kappa\in C^V$, and so $\phi^V$ when $ZFC+ORD\text{ is Mahlo}\vdash\phi\rightarrow\phi^V$. Therefore, if $M\vDash T$ then $V^M\vDash ZFC+ORD\text{ is Mahlo}$

I claim that this is equiconsistent with "$ORD$ is Mahlo" (This is not the same as $Ord$ being actually Mahlo, as I have explained elsewhere). It turns out "$ORD$ is Mahlo" is a natural limit point for these kind of Ackermann/$KM$ based theories. Let $T$ be your theory.

First, the easy part. The consistency strength of "$ORD$ is Mahlo" $\ge$ the consistency strength of $T$. Let $C=\{\alpha|V_\alpha\prec W\}$, where $W=\{x|x=x\}$. Note that if $\alpha\in C$ and $\phi(\alpha)$, then $\exists\beta(\beta\gt\alpha\land\phi(\beta))$. The reason for this is that else $\psi(\alpha)\leftrightarrow\beta\text{ is the largest } \beta\text{ such that }\phi(\beta)$ would be a definition of $\alpha$, and so $V_\alpha\vDash\exists x(\phi(x))$, which is a contradiction. Now, if $\phi(\alpha)$ with $\alpha\in C$, then then $\phi(\alpha\land\alpha\text{ is a cardinal})$ and so $V_\alpha\vDash\forall\eta(\exists\beta\gt\eta(\phi(\beta\land\beta\text{ is a cardinal}))$. Therefore, whenever $\kappa$ is inaccessible reflecting $V_\kappa\vDash T$, and the existsence of an inaccessible reflecting cardinal is equiconsistent with "$ORD$ is Mahlo."

Second, the hard part. The consistency strength of $T\ge$ the consistency strength of "$ORD$ is Mahlo." Let $C=\{\alpha|\phi(\alpha,p)\}$, and let $C^V=\{\alpha|\phi^V(\alpha,p)\}\cap Ord$ be club in $Ord$. Then you can see $\{\alpha|\phi^V(\alpha,p)\}$ is club (In the real class of ordinals); in particular $\phi^V(Ord,p)\land Ord\text{ is regular}$. Then we can find a non-empty class of regular $\kappa\in C^V$, and so $\phi^V$ when $ZFC+ORD\text{ is Mahlo}\vdash\phi\rightarrow\phi^V$. Therefore, if $M\vDash T$ then $V^M\vDash ZFC+ORD\text{ is Mahlo}$

I claim that this is equiconsistent with "$ORD$ is Mahlo" (This is not the same as $Ord$ being actually Mahlo, as I have explained elsewhere). It turns out "$ORD$ is Mahlo" is a natural limit point for these kind of Ackermann/$KM$ based theories. Let $T$ be your theory.

First, the easy part. The consistency strength of "$ORD$ is Mahlo" $\ge$ the consistency strength of $T$. Let $C=\{\alpha|V_\alpha\prec W\}$, where $W=\{x|x=x\}$. Note that if $\alpha\in C$ and $\phi(\alpha)$, then $\exists\beta(\beta\gt\alpha\land\phi(\beta))$. The reason for this is that else $\psi(\alpha)\leftrightarrow\beta\text{ is the largest } \beta\text{ such that }\phi(\beta)$ would be a definition of $\alpha$, and so $V_\alpha\vDash\exists x(\phi(x))$, which is a contradiction. Now, if $\phi(\alpha)$ with $\alpha\in C$, then $\phi(\alpha\land\alpha\text{ is a cardinal})$ and so $V_\alpha\vDash\forall\eta(\exists\beta\gt\eta(\phi(\beta\land\beta\text{ is a cardinal}))$. Therefore, whenever $\kappa$ is inaccessible reflecting $V_\kappa\vDash T$, and the existsence of an inaccessible reflecting cardinal is equiconsistent with "$ORD$ is Mahlo."

Second, the hard part. The consistency strength of $T\ge$ the consistency strength of "$ORD$ is Mahlo." Let $C=\{\alpha|\phi(\alpha,p)\}$, and let $C^V=\{\alpha|\phi^V(\alpha,p)\}\cap Ord$ be club in $Ord$. Then you can see $\{\alpha|\phi^V(\alpha,p)\}$ is club (In the real class of ordinals); in particular $\phi^V(Ord,p)\land Ord\text{ is regular}$. Then we can find a non-empty class of regular $\kappa\in C^V$, and so $\phi^V$ when $ZFC+ORD\text{ is Mahlo}\vdash\phi\rightarrow\phi^V$. Therefore, if $M\vDash T$ then $V^M\vDash ZFC+ORD\text{ is Mahlo}$