= Solution
Reflection in $L$ interchanges $x$ and $y$. Relabel the two points if necessary so that $x$ lies on the side of $L$ containing the center $a$; the absolute difference is symmetric in $x$ and $y$. Couple a Brownian motion $B$ started at $x$ to one $B'$ started at $y$ by setting $B'_t=\phi(B_t)$ before $T_L$ and $B'_t=B_t$ afterwards. The <Brownian reflection coupling> has the correct marginal laws because reflection is an isometry and the <Strong Markov property> applies at $T_L$. Once the paths meet on $L$, they agree forever.
If the path from $x$ does not hit the target circle before $T_L$, the coupled paths meet before the relevant uncoupled hitting outcomes can differ. The <coupling inequality for total variation> therefore gives
$$
\left|\mathbb P_x(B_{\tau_{a,r_1}}\in A)
-\mathbb P_y(B_{\tau_{a,r_1}}\in A)\right|
\leq\mathbb P_x(\tau_{a,r_1}<T_L).
$$
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