Where standards must be placed between studs, use hollow wall fasteners.
The shims have be long enough to be held in place by the wall fasteners.
Then attach the mounting bar to the 1-by-6 with lag screws (if they will penetrate the joists) or with sturdy hollow wall fasteners like molly bolts, and bolt the straps to the bar.
With the helper still steadying the cabinet, drill holes for screws and, where necessary, hollow wall fasteners through the mounting rails at the top and bottom of the cabinet.
Hollow wall fasteners should be equally sturdy.
Then either tighten the wall fasteners or fasten the frame to the wall with nails.
Rations or wall fasteners?
Because they are heavy and fragile, caution is needed when handling them, and wall fasteners must be installed carefully.
Because mirrors are heavy, awkward and fragile, more planning and organization are necessary, and greater attention must be given to installing the wall fasteners.