Looking more carefully at the above McMullen's paper, I realized that the question has the positive answer due to Theorem 3 in the paper.

McMullen constructs homomorphisms $\Pi'_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on some element, then the element vanishes. Let $u$ be a non-zero linear functional. For a polytope $P$ denote $$P_u:=\{x\in P|\, u(x)=\max_{y\in P}u(y).\}$$ Let us define the homomorphism $f_u\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ on generators by $f_u([P]):=vol(P_u)$ where $vol$ is $(n-1)$-dimensional volume. McMullen shows that $f_u$ is well defined.

This construction can be generalized recursively. Let $u_1,\dots,u_k$ be linearly independent linear functionals. Define $P_{u_1,\dots,u_k}:=(P_{u_{1},\dots,u_{k-1}})_{u_k}$. Define the homomorphism $f_{u_1,\dots,u_k}\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ by $$f_{u_1,\dots,u_k}([P]):=vol(P_{u_{1},\dots,u_{k}}),$$ where $vol$ is $(n-k)$-dimensional volume.

Theorem 3.(McMullen) All the homomorphisms $f_{u_1,\dots u_k}$ separate points in $ \Pi'_{\mathbb{F}}$.

Let us show how Theorem 3 answers (immediately) the question in the post. Let $w\in\Pi'_{\mathbb{Q}} $ belongs to the kernel of the homomorphism in the question. That implies that for all $k$-tuples of linearly $\mathbb{R}$-independent real linear functionals $u_1,\dots,u_k$ one has $f_{u_1,\dots,u_k}(w)=0$. Now if $u_1,\dots,u_k$ are rational linear functionals which are linearly independent over $\mathbb{R}$ then they are linearly independent over $\mathbb{Q}$. For all of such $k$-tuples all $f_{u_1,\dots ,u_k}$ separate points in $\Pi'_{\mathbb{Q}}$. Hence $w=0$.

QED.

Looking more carefully at the above McMullen's paper, I realized that the question has the positive answer due to Theorem 3 in the paper.

McMullen constructs homomorphisms $\Pi'_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on some element, then the element vanishes. Let $u$ be a non-zero linear functional. For a polytope $P$ denote $$P_u:=\{x\in P|\, u(x)=\max_{y\in P}u(y).\}$$ Let us define the homomorphism $f_u\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ on generators by $f_u([P]):=vol(P_u)$ where $vol$ is $(n-1)$-dimensional volume. McMullen shows that $f_u$ is well defined.

This construction can be generalized recursively. Let $u_1,\dots,u_k$ be linearly independent linear functionals. Define $P_{u_1,\dots,u_k}:=(P_{u_{1},\dots,u_{k-1}})_{u_k}$. Define the homomorphism $f_{u_1,\dots,u_k}\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ by $$f_{u_1,\dots,u_k}([P]):=vol(P_{u_{1},\dots,u_{k}}),$$ where $vol$ is $(n-k)$-dimensional volume.

Theorem 3.(McMullen) All the homomorphisms $f_{u_1,\dots u_k}$ separate points in $ \Pi'_{\mathbb{F}}$, i.e. if on some element $w\in \Pi'_{\mathbb{F}} $ all such homomorphisms vanish then $w=0$.

Let us show how Theorem 3 answers (immediately) the question in the post. Let $w\in\Pi'_{\mathbb{Q}} $ belongs to the kernel of the homomorphism in the question. That implies that for all $k$-tuples of linearly $\mathbb{R}$-independent real linear functionals $u_1,\dots,u_k$ one has $f_{u_1,\dots,u_k}(w)=0$. Now if $u_1,\dots,u_k$ are rational linear functionals which are linearly independent over $\mathbb{R}$ then they are linearly independent over $\mathbb{Q}$. For all of such $k$-tuples all $f_{u_1,\dots ,u_k}$ separate points in $\Pi'_{\mathbb{Q}}$. Hence $w=0$.

QED.

Looking more carefully at the above McMullen's paper, I realized that the question has a positive answer by Theorem 3 in the paper.

McMullen constructs homomorphisms $\Pi'_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on some element, then the element vanishes. Let $u$ be a non-zero linear functional. For a polytope $P$ denote $$P_u:=\{x\in P|\, u(x)=\max_{y\in P}u(y).\}$$ Let us define the homomorphism $f_u\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ on generators by $f_u([P]):=vol(P_u)$ where $vol$ is $(n-1)$-dimensional volume. McMullen shows that $f_u$ is well defined.

This construction can be generalized recursively. Let $u_1,\dots,u_k$ be linearly independent linear functionals. Define $P_{u_1,\dots,u_k}:=(P_{u_{1},\dots,u_{k-1}})_{u_k}$. Define the homomorphism $f_{u_1,\dots,u_k}\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ by $$f_{u_1,\dots,u_k}([P]):=vol(P_{u_{1},\dots,u_{k}}),$$ where $vol$ is $(n-k)$-dimensional volume.

Theorem 3.(McMullen) All the homomorphisms $f_{u_1,\dots u_k}$ separate points in $ \Pi'_{\mathbb{F}}$.

Let us show how Theorem 3 answers (immediately) the question in the post. Let $w\in\Pi'_{\mathbb{Q}} $ belongs to the kernel of the homomorphism in the question. That implies that for all $k$-tuples of linearly $\mathbb{R}$-independent real linear functionals $u_1,\dots,u_k$ one has $f_{u_1,\dots,u_k}(w)=0$. Now if $u_1,\dots,u_k$ are rational linear functionals which are linearly independent over $\mathbb{R}$ then they are linearly independent over $\mathbb{Q}$. For all of such $k$-tuples all $f_{u_1,\dots ,u_k}$ separate points in $\Pi'_{\mathbb{Q}}$. Hence $w=0$.

QED.

Looking more carefully at the above McMullen's paper, I realized that the question has the positive answer due to Theorem 3 in the paper.

McMullen constructs homomorphisms $\Pi'_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on some element, then the element vanishes. Let $u$ be a non-zero linear functional. For a polytope $P$ denote $$P_u:=\{x\in P|\, u(x)=\max_{y\in P}u(y).\}$$ Let us define the homomorphism $f_u\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ on generators by $f_u([P]):=vol(P_u)$ where $vol$ is $(n-1)$-dimensional volume. McMullen shows that $f_u$ is well defined.

This construction can be generalized recursively. Let $u_1,\dots,u_k$ be linearly independent linear functionals. Define $P_{u_1,\dots,u_k}:=(P_{u_{1},\dots,u_{k-1}})_{u_k}$. Define the homomorphism $f_{u_1,\dots,u_k}\colon \Pi'_{\mathbb{F}}\to \mathbb{F}$ by $$f_{u_1,\dots,u_k}([P]):=vol(P_{u_{1},\dots,u_{k}}),$$ where $vol$ is $(n-k)$-dimensional volume.

Theorem 3.(McMullen) All the homomorphisms $f_{u_1,\dots u_k}$ separate points in $ \Pi'_{\mathbb{F}}$.

Let us show how Theorem 3 answers (immediately) the question in the post. Let $w\in\Pi'_{\mathbb{Q}} $ belongs to the kernel of the homomorphism in the question. That implies that for all $k$-tuples of linearly $\mathbb{R}$-independent real linear functionals $u_1,\dots,u_k$ one has $f_{u_1,\dots,u_k}(w)=0$. Now if $u_1,\dots,u_k$ are rational linear functionals which are linearly independent over $\mathbb{R}$ then they are linearly independent over $\mathbb{Q}$. For all of such $k$-tuples all $f_{u_1,\dots ,u_k}$ separate points in $\Pi'_{\mathbb{Q}}$. Hence $w=0$.

QED.