In the literature of mechanism design, market mechanisms have been developed by professionals based on their experience. The concept of automated mechanism design (AMD), initiated by Sandholm (2002), is a ground-breaking computer-aided framework to develop market mechanisms. In this paper, we apply a very recent AMD approach based on Boolean Satisfiability (SAT) to the mechanism design of false-name-proof facility location. We first provide a general theoretical characteristic of false-name-proof mechanisms, which enables a quite compact representation of target mechanisms. Our approach successfully reproduces several known results in the literature on false-name-proof facility locations over discrete structures. Furthermore, some unknown mechanisms are discovered for locating a public good on a 2-by-2 grid, and an impossibility result is revealed for locating a public bad, with an additional mild assumption, on a 2-by-3 grid. Finally, we demonstrate the extendability of our approach, by providing a new false-name-proof mechanism for a slightly modified problem of locating a public good.