That’s similar to my general approach, and it is a pretty darned solid way to handle fist-sized holes. Unfortunately, from what it sounds like there isn’t even enough space between the sheetrock and the panel to slide a dime between them.
Anyway, for the benefit of other home gamers, when I have done this kind of thing I usually have a big hole and a square of drywall to fill it. I put one or two long furring strips inside the hole using drywall screws
beyond the edges of the hole to hold the strips in place–glue is optional. This leaves me with a couple of nice wood crosspieces to which I screw the piece of new drywall. Then standard taping and finishing.
I avoid spackle (the stuff one uses to patch nail holes) and go directly to joint compound or plaster. Given enough patience it will be impossible for anyone to tell where the repair was made.
I enjoy taping and working with joint compound because it makes such a huge difference in the appearance when a good job is done.
An easy way I have found but still a little work is to cut say a 3" by 3" square hole in the drywall. Now cut a 2 7/8" by 6" piece of 1/2" plywood. Use a string to insert the plywood into the hole and pull it tight overlapping about equally on both sides. A liquid nail or something similar should have been spread on the area that would be overlapped. I usually preset a screw so the plywood patch is allready threaded and easy to reinstall the screw once placed. The screw in the middle is needed to hold the plywood in place while it is screwed in to the existing drywall. Now just fill in the hole with some drywall and use drywall patch. You can also just ise another pice of plywood instead of drywall and float over it. Just make sure it is thinner than your drywall.