Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. In this paper we consider the case when all variables occurring below such function symbols are quantified over a finite subset of their domains. We present a non-naive decision procedure for background theories extended this way on top of black-box decision procedures for the EA-fragment of the background theory. In its core, it employs a model-guided instantiation strategy for obtaining pure background formulas that are equi-satisfiable with the original formula. Unlike traditional finite model finders, it avoids exhaustive instantiation and, hence, is expected to scale better with the size of the domains. Our main results in this paper are a correctness proof and first experimental results.