= Solution
For every site, $P(R(x)\geq r)=\beta_r$. This event depends only on <edges> with both endpoints in $x+\Lambda_r$: any connecting path can be stopped on its first hit of that boundary. Since $\lambda>0$, $\beta_r\to0$, so all clusters have finite radius almost surely, by a countable union over sites. Set the radius of an isolated site to zero.
Write $a_n=(d/\lambda)\log n$. It suffices to take $0<\epsilon<1$, since larger error intervals contain one such interval. For any small $\delta>0$, the decay-rate limit gives, for all large $r$,
$$
e^{-(\lambda+\delta)r}\leq\beta_r\leq e^{-(\lambda-\delta)r}.
$$
For the upper tail use $r_+=\lceil(1+\epsilon)a_n\rceil$ and a <union bound>:
$$
P(M_n\geq r_+)\leq(2n+1)^d\beta_{r_+}
\leq Cn^{d-(\lambda-\delta)(1+\epsilon)d/\lambda}\longrightarrow0,
$$
provided $\delta<\lambda\epsilon/(1+\epsilon)$.
For the lower tail let $r_-=\lfloor(1-\epsilon)a_n\rfloor+1$. Choose sites in $\Lambda_{n-r_-}$ spaced by $2r_-+1$ in each coordinate. Their radius-$r_-$ boxes are vertex-disjoint, so their local connection events are independent. There are $N_n\geq c(n/r_-)^d$ such sites for large $n$. If $M_n\leq(1-\epsilon)a_n$, none of these events occurs. Thus
$$
P(M_n\leq(1-\epsilon)a_n)\leq(1-\beta_{r_-})^{N_n}\leq e^{-N_n\beta_{r_-}},
$$
where
$$
N_n\beta_{r_-}\geq\frac{c}{(\log n)^d}
n^{d-(\lambda+\delta)(1-\epsilon)d/\lambda}\longrightarrow\infty
$$
if $\delta<\lambda\epsilon/(1-\epsilon)$. Choose one $\delta$ satisfying both restrictions. The integer choices give the required strict inequalities. Thus the <maximum cluster radius under exponential one-arm decay> has <convergence in probability>:
$$
\boxed{\frac{M_n}{(d/\lambda)\log n}\longrightarrow1\quad\text{in probability}}.
$$
The logarithmic box-packing loss does not change the leading constant $d/\lambda$.
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