Is there any problem in $\mathsf{\Sigma^P_2}$ which is solvable in bounded tree width graphs? I'm looking for a problem which belongs to $\mathsf{\Sigma^P_2}$ in general graphs but is in $\mathsf{P}$ in bounded tree width graphs, In fact I think this problems are harder than using normal dynamic programming in bounded-treewidth graphs to solve them.
I think 2-clique-coloring [GT19 in Schaefer and Umans] is an example. The question is whether the given graph can be (improperly) 2-colored in such a way that none of its maximal cliques are monochromatic. For graphs of bounded treewidth, each maximal clique should occur within a single bag of the tree decomposition, so it should work to use the standard dynamic programming approach in which the states of the dynamic program are 2-colorings of the bag that correctly color all maximal cliques within the bag and are consistent with good states of the child bags.