$\newcommand\R{\mathbb R}$Here is a plan for any dimension $m$.
Without loss of generality, we have two open balls: $B_1=B_0(r_1)$ and $B_2=B_D(r_2)$, for some $D\in\R^m$ with $\|D\|=d$, where $B_x(r)$ is the open ball in $\R^m$ of radius $r$ centered at $x$ and $\|\cdot\|$ is the Euclidean norm.
Take any unit vector $u$. Take any $x\in B_1$ such that $x\cdot u=0$, where $\cdot$ denotes the dot product. Then the condition that the line through $x$ in the direction of $u$ intersects $B_2$ is that $x\in B_D(R_u)$, where $R_u:=\sqrt{(D\cdot u)^2+r_2^2}$; this can be obtained by minimizing $\|x-tu-D\|^2$ in real $t$ and taking into account the condition $x\cdot u=0$.
So, $$\mathcal A(C,u)=L_{m-1}(B_0(r_1)\cap B_D(R_u)\cap H_u),$$ where $H_u:=\{x\in\R^m\colon\, x\cdot u=0\}$ and $L_{m-1}$ is the $(m-1)$-dimensional Lebesgue measure over the hyperplane $H_u$.
So, $\mathcal A(C,u)$ is the $(m-1)$-volume of the intersection of two balls, $B_0(r_1)\cap H_u$ and $B_D(R_u)\cap H_u$, in $H_u$. So, $\mathcal A(C,u)$ is an ordinary integral of the ($(m-2)$-dimensional) volumes of $(m-2)$-dimensional balls, which is actually known -- but is expressed via an incomplete beta function, rather than in elementary functions.
After that, it will remain to integrate $\mathcal A(C,u)$ over all $u$ on the $(m-1)$-dimensional unit sphere.