= Solution
Use the coordinatewise <partial order>: $\omega\leq\eta$ means $\omega(e)\leq\eta(e)$ for every $e\in E$. An <increasing event> $A$ is upward closed: $\omega\in A$ and $\omega\leq\eta$ imply $\eta\in A$.
Under the independent <Bernoulli distribution> <product measure> $\mathbb P_p$ on these coordinates, the <Harris-FKG inequality> states
$$
\boxed{\mathbb P_p(A\cap B)\geq\mathbb P_p(A)\mathbb P_p(B)}
$$
for <increasing events> $A,B$. For $K\subseteq E$, let $[\omega]_K$ be the cylinder of configurations agreeing with $\omega$ on $K$. The <disjoint occurrence of increasing events> $A\square B$ consists of configurations for which there exist disjoint $K,L$ with $[\omega]_K\subseteq A$ and $[\omega]_L\subseteq B$. For <increasing events>, witnesses can be taken to prescribe only open coordinates. The <BK inequality> is
$$
\boxed{\mathbb P_p(A\square B)\leq\mathbb P_p(A)\mathbb P_p(B).}
$$
Positive association rewards simultaneous occurrence; the disjoint-witness requirement in the <BK inequality> gives a bound in the opposite direction. The inequalities also apply to different independent Bernoulli parameters in different coordinates.
Regard the boxes as sets of lattice <graph vertices>. Write $\tau_p(x,y)=\mathbb P_p(x\leftrightarrow y)$, and put $g_0=1$. On the event defining $g_n$, choose an open <self-avoiding walk> from the origin to its first visit to $\partial\Lambda_n$. Let $y$ be its first visit to $\partial\Lambda_m$. Its initial segment witnesses $0\leftrightarrow y$. Its remaining segment ends at a point $z$ with
$$
\|z-y\|_\infty\geq\|z\|_\infty-\|y\|_\infty=n-m.
$$
For $m<n$, truncate this remaining segment on its first visit to $y+\partial\Lambda_{n-m}$. The two segments use disjoint <edges>, so they are disjoint witnesses. For $m=n$, the second event is the sure event with empty witness. Thus the <union bound>, the <BK inequality> and translation invariance give
$$
\begin{aligned}
g_n&\leq\sum_{y\in\partial\Lambda_m}
\mathbb P_p\bigl(\{0\leftrightarrow y\}\square
\{y\leftrightarrow y+\partial\Lambda_{n-m}\}\bigr)\\
&\leq\boxed{g_{n-m}\sum_{y\in\partial\Lambda_m}\tau_p(0,y)}.
\end{aligned}
$$
An unrestricted connection event can depend on infinitely many <edges>. Apply the finite-coordinate <BK inequality> first to connections confined to growing boxes and take increasing limits; each occurrence has a finite path witness. This justifies its use here. The estimate is the <weighted BK boundary-splitting estimate>.
By summing cluster indicators and using <Tonelli theorem>, the <percolation susceptibility> is
$$
\chi(p)=\mathbb E_p|C(0)|=\sum_{y\in\mathbb Z^d}\tau_p(0,y)
=1+\sum_{m\geq1}S_m,\qquad
S_m=\sum_{y\in\partial\Lambda_m}\tau_p(0,y).
$$
If $\chi(p)<\infty$, then $S_m\to0$. At $p=0$ all $g_k$ vanish, so any positive exponential rate works. Otherwise choose $m\geq1$ with $a=S_m\in(0,1)$. Iterating the <weighted BK boundary-splitting estimate> gives, for $k=jm+r$ with $0\leq r<m$,
$$
g_k\leq a^j g_r\leq a^{\lfloor k/m\rfloor}.
$$
For $k\geq m$, $\lfloor k/m\rfloor\geq k/(2m)$, so $g_k\leq\exp[-(-\log a)k/(2m)]$. To include the finitely many smaller indices without a prefactor, note that $p<1$ under finite susceptibility and
$$
g_k\leq g_1=1-(1-p)^{2d}<1\qquad(k\geq1).
$$
Therefore one may take
$$
\boxed{\mu(p)=\min\left\{\frac{-\log a}{2m},
\frac{-\log g_1}{m}\right\}>0,\qquad
g_k\leq e^{-\mu(p)k}\quad(k\geq1).}
$$
For $k<m$ this follows from $\mu(p)k\leq-\log g_1$, and for $k\geq m$ it follows from the iterated estimate. This proves <exponential one-arm decay from finite susceptibility> with exactly the requested unit prefactor.
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