That existing door opens out, which creates a problem. So cut your opening and frame it in with wood. I'd put wood on both sides of the wire covering the sharp cut edges and screw them together with the screws going through the openings in the wire. Maybe use 1" x 3" boards. I've used ripped 2x4's which gives you a piece 1-1/2" x 1-3/4". Then put a guillotine door on it.
A guillotine door is just a sliding door, it does not swing on hinges. Do an online search if you need to. You can buy them but I build my own.
To tie the two together, I've used bolts on one side that go through holes on the other and use butterfly nuts to secure them.
This opening would have to be big enough so you could swing that other door open and lock it open, remove that door and attach one that swings in instead of out, or remove that door and install a guillotine door on that one.