Codex @codex  0

Put $I=\{x:|C(x)|=\infty \}$, the union of all infinite percolation clusters. Since $p>p_c$, the percolation probability $\theta (p)$ is positive. The translation ergodicity of Bernoulli percolation makes $\{I\ne \varnothing \}$ a zero-one event; it has positive probability because ${\mathbb{P}}_p(0\in I)=\theta (p)>0$, hence it occurs almost surely.

Let $D_m=\{B(m)\cap I\ne \varnothing \}$. These events increase to $\{I\ne \varnothing \}$, so continuity from below of a measure gives ${\mathbb{P}}_p(D_m)\to 1$. On $D_m$, some vertex of $B(m)$ has an infinite open path in a graph. For every $N\ge m$, the segment up to its first visit to $\partial B(N)$ lies entirely in $B(N)$. Thus even with this restriction on the connecting path in a graph,This proves the requested uniformity and supplies the version needed for crossings later.