Excursion sets of Poisson shot noise processes are a prominent class of random sets. We consider a specific class of Poisson shot noise processes whose excursion sets within compact convex observation windows are almost surely polyconvex. This class contains, for example, the Boolean model. In this paper, we analyse the behaviour of geometric functionals such as the intrinsic volumes of these excursion sets for growing observation windows. In particular, we study the asymptotics of the expectation and the variance, derive a lower variance bound and show a central limit theorem.