= Solution
Fix $\epsilon>0$ and choose a cylinder event $B$ with $\mathbb P_p(A\mathbin\triangle B)<\epsilon$. Translate $B$ far enough that $B$ and $\phi(B)$ depend on disjoint edge sets. They are independent, while automorphism invariance gives $\phi(A)=A$. The symmetric-difference inclusion supplied in the question gives
$$
\left|\mathbb P_p(A)-\mathbb P_p(B\cap\phi(B))\right|\leq2\epsilon.
$$
Also $|\mathbb P_p(A)-\mathbb P_p(B)|<\epsilon$, and independence gives $\mathbb P_p(B\cap\phi(B))=\mathbb P_p(B)^2$. Letting $\epsilon\downarrow0$ yields
$$
\mathbb P_p(A)=\mathbb P_p(A)^2,
$$
so $\mathbb P_p(A)\in\{0,1\}$.
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