Observe that $e$ sends an element $\downarrow x$ to an element of the form $\downarrow y$, and an element $\downarrow x'$ to an element of the form $\downarrow y'$. Now, if $x'$ and $y'$ are distinct in $Q$, and if $x,y$ are such that $e(x)=x'$ and $e(y)=y'$, then there exists a finite intersection $U$ of the elements of the sub base you gave above, and a finite intersection $V$ of these elements, such that $U\cap V = \emptyset$. Let us denote $U=u_1\cap u_2\cap\ldots\cap u_k$, with $u_i = P-s_i$ and $s_i=\downarrow \alpha$ or $s_i=\uparrow \alpha$, and $V = v_1\cap \ldots v_r$, with $v_j = P - t_j$ and $ t_j=\downarrow \beta$ or $t_i = \uparrow \beta$. Then by booleanity, $P=\cup_i s_i \cup \cup_j t_j$. It follows that $e(P)=Q=\cup_i e(s_i) \cup \cup_j e(t_j)$, and conclude by booleanity that $Q$ is Hausdorff.

Observe that $e$ sends an element $\downarrow \alpha$ to an element of the form $\downarrow \alpha'$, and an element $\downarrow \beta$ to an element of the form $\downarrow \beta'$. Now, if $x'$ and $y'$ are distinct in $Q$, and if $x,y$ are such that $e(x)=x'$ and $e(y)=y'$, then there exists a finite intersection $U$ of the elements of the sub base you gave above, and a finite intersection $V$ of these elements, such that $U\cap V = \emptyset$. Let us denote $U=u_1\cap u_2\cap\ldots\cap u_k$, with $u_i = P-s_i$ and $s_i=\downarrow \alpha$ or $s_i=\uparrow \alpha$, and $V = v_1\cap \ldots v_r$, with $v_j = P - t_j$ and $ t_j=\downarrow \beta$ or $t_i = \uparrow \beta$. Then by booleanity, $P=\cup_i s_i \cup \cup_j t_j$. It follows that $e(P)=Q=\cup_i e(s_i) \cup \cup_j e(t_j)$, and conclude by booleanity that $Q$ is Hausdorff.