We consider the shallow water magnetohydrodynamic (SWMHD) equations, in which multiscale phenomena appear in geophysics, astrophysics, and energy production applications. However, capturing both large and small scales numerically requires an exact treatment of the physically-observed divergence-free condition of the magnetic field and typically a very fine spatial grid. An alternative to requiring a fine spatial grid is to use a well-balanced scheme, which preserves the steady-state solution exactly, allowing one to capture multiscale phenomena on a less computationally-expensive grid. We propose a well-balanced scheme using the flux globalization approach that does exactly that for the SWMHD balance laws, along with preserving the divergence-free condition of the magnetic field exactly. The designed scheme is successfully tested on a number of examples.