= Solution
At the meeting time,
$$
A_T^+=A_T^-=\frac{C_T}{\sqrt2}.
$$
The process $C$ is independent of $T$, so conditional on $T=u$ the meeting position is $N(0,u/2)$. Independently, $B_t-B_u$ is $N(0,t-u)$. Therefore
$$
Z_t\mid\{T=u\}\sim N(0,t-u/2).
$$
For $0<s\leq t$, integrate this conditional <Gaussian distribution> against the density from part c:
$$
\boxed{
\mathbb P(T\leq s,\ Z_t\leq z)
=\int_0^s
\Phi\left(\frac z{\sqrt{t-u/2}}\right)
\frac a{\sqrt{\pi u^3}}e^{-a^2/u}\,du.}
$$
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