  We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral
homology $S^1 \times S^3$ along an embedded torus, which could either be
regarded as an extension of the product formula for Seiberg-Witten invariants
or a manifestation of the surgery exact triangle in $4$-dimensional
Seiberg-Witten theory of homology $S^1 \times S^3$. As an application, we
compute this invariant for mapping tori of $3$-manifolds under diffeomorphisms
of finite order and fixed-point set being a simple closed curve. This
computation generalizes the result of Lin-Ruberman-Saveliev.
