Dave
Making it tilt proof is the easiest part of the project. Just get some sort of a strap ( old shoulder strap you can cut up or short piece of small chain ) screw it to the back of the case and find a spot in the wall with a stud. Screw it in there and your done. One small hole in the wall, no big deal.
Another way would be to get an L bracket and screw that to the bottom of the case and then to the wall.
You can also get a commercially made tray style hard case but that will probably go for about $175 or more. NY Case Company does custom cases.