Three different minus signs

The reduced graviton Hessian is $-\square \mathcal{P}$, so its inverse is $-\mathcal{P}{{\mathsf{E}}_0}^{\pm }$.

The scalar action Hessian is $-(\square +m^2)$, so its inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$, even though the equation inverse is $+{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$.

The retarded-minus-advanced scalar Hessian kernel is therefore $-{\delta }_m$. With the convention (eq:CCR),

Since ${\partial }_{x^0}{\delta }_m$ at equal times is the spatial delta, this gives $[\varphi ,{\partial }_t\varphi ]=i\hslash {\delta }_{{\mathbb{R}}^3}$ in the canonical ordering. Thus the action and commutator signs agree.