How should I support this large drywall patch?
It's a matter of preference, but I would float scrap lumber backing at four locations:


Down both sides
Across the bottom between the two studs
Across the between the two studs just below the plumbing penetrations
Anywhere else that seems too flexy when you do a little press-testing


1/2" or thicker plywood and 1-by or 2-by lumber work well. This backing doesn't really need to be attached to the studs in all cases (doing so can make it difficult to keep everything flush), but should be fastened well to the surrounding drywall. Construction adhesive would reduce the number of screws necessary there. 

Protip: If you don't have access to a dedicated screw gun (which is not the same as a drill, despite common usage of the term), these drywall screw setter bits are golden because they keep the screw from sinking too far into the drywall: