= Solution
At fixed time, the coordinate field and the <canonical momentum> are independent variables. Their <functional derivatives> satisfy
$$
\frac{\delta\phi_a(\mathbf x)}{\delta\phi_b(\mathbf y)}=\delta_{ab}\delta^{(3)}(\mathbf x-\mathbf y),\qquad\frac{\delta\phi_a(\mathbf x)}{\delta\pi_b(\mathbf y)}=0.
$$
Putting $F=Q$, $G=\phi_a(\mathbf x)$ in the <Poisson bracket> definition therefore gives
$$
\{Q,\phi_a(\mathbf x)\}_{\rm PB}=-\int d^3y\,\frac{\delta Q}{\delta\pi_b(\mathbf y)}\delta_{ba}\delta^{(3)}(\mathbf x-\mathbf y),\qquad\boxed{\{Q,\phi_a(x)\}_{\rm PB}=-\frac{\delta Q}{\delta\pi_a(x)}.}
$$
The superscript on the species label $a$ in the printed denominator has no spacetime-metric significance; with the ordinary species Kronecker metric it is the same expression.
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