= Solution
For $\delta x^0=\epsilon^0$ the <Noether charge> is $Q=-\epsilon^0H$, with the <Hamiltonian density> $\mathcal H=\pi_a v_a-\mathcal L(\phi,v,\nabla\phi)$ and $v_a=\dot\phi_a$. Invertibility of the momentum-velocity relation allows $v_a=v_a(\phi,\pi,\nabla\phi)$. When taking a <functional derivative> at fixed $\phi$, the implicit dependence of $v$ cancels in the <Legendre transform>:
$$
\frac{\delta H}{\delta\pi_a}=v_a+\left(\pi_b-\frac{\partial\mathcal L}{\partial v_b}\right)\frac{\partial v_b}{\partial\pi_a}=v_a.
$$
It follows that
$$
\boxed{\{\phi_a,Q\}_{\rm PB}=\frac{\delta Q}{\delta\pi_a}=-\epsilon^0\dot\phi_a=\bar\delta\phi_a.}
$$
This proves the same generator relation for time translations. In a constrained theory the assumed inversion fails, so the appropriate reduced variables or <Dirac brackets> are needed instead. That is precisely the issue encountered for the <Proca field> in Question 4.
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