Tree automata and pigeonhole classes of matroids -- II

Let $\psi$ be a sentence in the counting monadic second-order logic of matroids. Let F be a finite field. Hlineny's Theorem says there is a fixed-parameter tractable algorithm for testing whether F-representable matroids satisfy $\psi$, with respect to the parameter of branch-width. In a previous paper we proved there is a similar fixed-parameter tractable algorithm for any efficiently pigeonhole class. In this sequel we apply results from the first paper and thereby extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, when H is a finite group. As a consequence, we can obtain a new proof of Courcelle's Theorem.