Ross, have you considered embedding some rare earth magnets on the side edges of the back panel that align with matching ones in the insides of the side panels? The panel would snap into place when you rotated the panel vertically. (I’m assuming the use of the little nubs on the bottom edge you mentioned.) If this idea actually works, it would be completely invisible from any viewing angle.
It looks like your back panel is fairly thick, so I bet you could use 3/8” diameter magnets. Some of these magnets are really strong. Maybe you could get away with one magnet on each side edge of the back panel near the top. If that’s not strong enough, you could add another pair a third of the way down. Then you could go down another third if necessary.
Regardless of what you do, won’t you also need a small pull (tastefully designed with a small projection, of course) or routed finger pull cavity near the top so you can rotate the panel down? The fit of your panel is so precise that it will be difficult to start opening otherwise.