 Solution

Fix $ϵ>0$ and choose a cylinder event $B$ with ${\mathbb{P}}_p(A△B)<ϵ$. Translate $B$ far enough that $B$ and $\varphi (B)$ depend on disjoint edge sets. They are independent, while automorphism invariance gives $\varphi (A)=A$. The symmetric-difference inclusion supplied in the question givesAlso $|{\mathbb{P}}_p(A)-{\mathbb{P}}_p(B)|<ϵ$, and independence gives ${\mathbb{P}}_p(B\cap \varphi (B))={\mathbb{P}}_p(B{)}^2$. Letting $ϵ\downarrow 0$ yieldsso ${\mathbb{P}}_p(A)\in \{0,1\}$.