It is generally true that if $(X,\mathcal{X})$ is a measurable space, $(Y,\mathcal{Y})$ a standard Borel space, and $\kappa:X\to\mathcal{P}(X)$ a transition probability, then the measure-valued function that maps $x$ to the atomic part of $\kappa(x)$ is measurable. The rest is a red herring.
With the usual $\sigma$-algebra on $\mathcal{P}(X)$ (the space of probability measures on $X$) generated by evaluation of measurable sets, the function $j:\mathcal{P}(X)\times X\to [0,1]$ given by $j(\mu,x)=\mu\big(\{x\}\big)$ is measurable. You can find the details in Appendix A of "Fundamentals of Nonparametric Bayesian Inference" (2017) by Ghosal and van der Vaart. Let $C\subseteq\mathcal{P}(X)\times X$ be the set on which $j$ is strictly positive. Let $g_\mu$ be the $\mu$-section of $C$. By a standard generalization of Fubini's theorem for transition probabilities, the function $\mu\mapsto \int g_\mu~\mathrm d\mu$ is measurable and so is the function $\mu\mapsto \int_A g_\mu~\mathrm d\mu$, which gives you the desired measurability.
For another approach, look at Theorem 2.12 of
Dubins, Lester, and David Freedman. "Measurable sets of measures." Pacific Journal of Mathematics 14.4 (1964): 1211-1222.