You can also try a "plaque"-type mount...mounting a larger piece of wood that'll span 2 studs, and mounting the FMB through the "plaque". (you can paint the "plaque" the same color as the walls)

Or a shelf (hanging from the studs) and then hanging the FMB from the bottom of the shelf (to allow use of the entire shelf).

Or, as mentioned, cut out the drywall, insert a brace, put the drywall back, mud it, paint it.

But don't count on the drywall for support.