I follow the reasoning from your previous question.
One contribution to $1-P_{A\rightarrow B}$ is that the event $A$ does not happen at all in a time $T$, that probability is $1-P_A(T)$. For the remaining contributions to $1-P_{A\rightarrow B}$ the event $A$ happens at least once, denote by $\tau$ the time at which this first happens, with probability density $dP_A/d\tau$. Then from $\tau$ to $T$ the event $B$ does not happen, with probability $e^{-(T-\tau)\lambda}$. This gives in total $$1-P_{A\rightarrow B}=1-P_A(T)+\int_{0}^T \frac{dP_A(\tau)}{d\tau}e^{-(T-\tau)\lambda}\,d\tau.$$