= Solution
Write
$$
H_A(z)=\mathbb P_z(B_{\tau_{a,r_1}}\in A).
$$
This is the <harmonic measure> of $A$ viewed from $z$, and is a bounded <harmonic function> outside the closed target disc. The disc is contained in $B(0,r_2)$. In the half-plane cut out by a line $L$ through the origin, the probability of reaching $B(0,r_2)$ before $L$, from a point of modulus $R$, is $O(r_2/R)$ uniformly in the direction. One sees this by mapping the half-plane outside the disc with $z\mapsto\log(z/r_2)$ to a half-strip and solving the corresponding <Dirichlet problem> by a sine series.
Part (b) consequently shows that the angular oscillation of $H_A$ on a large circle tends to zero, uniformly in the Borel set $A$. The same half-strip estimate in the annulus $R-|a|\leq|z|\leq R+|a|$ shows that shifting the center of that circle by the fixed vector $a$ changes the average of $H_A$ by $o(1)$, uniformly in $A$. Hence, for every $\varepsilon>0$, all sufficiently large $R$ satisfy
$$
\left|\mathbb P_{\nu_{a,R}}(B_{\tau_{a,r_1}}\in A)
-\mathbb P_{\nu_{0,R}}(B_{\tau_{a,r_1}}\in A)\right|
\leq\varepsilon
$$
for every Borel subset $A$ of the target disc.
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