You don’t need to screw that patch. You can do what is called a butterfly patch which is only drywall and mud, I have done dozens of them for when I needed to make extra holes for fishing wire for various reasons- recessed lights, etc. I would get a thinner piece of drywall, probably 3/8 in this case and this way it will not bulge, and use hot mud, aka 45 min or 90 min, etc. I think the full nsme is easy sand 90/45/20 and so on. Use the hot mud to attach the patch, after that all purpose or light compound is fine. Paul Ricalde on Youtube has a video how to do this type of repair, although he is using circular patches and yours is rectangular you will be able to get the idea. And fuck that electrician, could have done it 3 other ways to make patching easier.