Your gym has bumper plates, matting and platforms, but still allows no o-lifts or even OH press (which isn't really an o-lift)? Very bizarre. Seems like a total waste and dumb management to me. Any chance you can change gyms?
SociallyChallenged's idea of handstand push-ups (HSPU) as a substitute is a good one, here are a few things to consider:
1. The range of motion in an OH press is greater than in an HSPU, as you can't lower your body past the top of your head in the HSPU but the bar goes all the way to your shoulders in the OH press. You can solve this by buying or making some paralletes, however.
2. It's much harder to progressively load the HSPU. For one thing, with the OH press, you start with a bare bar and work your way up. With HSPU, you start with your BW, which you might not be able to do right away. If that's the case, you can start with an easier version of the HSPU - check out the "Handstand push-up variations" video on the
CF exercises page for some ideas. However, once you can handle your BW for 5x5, it then becomes difficult to increase difficulty. I suppose you could get weighted vests and load them up with various loads, but it won't be anywhere as easy as just adding more weight to the bar for OH press.
3. The job of balancing a weight above your head in the OH press is one of the big benefits of the exercise, as it strengthens your entire upper body (especially "the core") and not just the shoulders. While the HSPU definitely requires a ton of balance & coordination, the actual amount of strength required to keep yourself upright becomes a smaller and smaller part of the exercise as you get better at it, until the only real limiting factor is the strength of your shoulders/traps/triceps. It's still a fantastic exercise, but its value to your core diminishes as your coordination improves.
In short, if you can find any way to do the OH press - becoming friends with the employees, doing it at off hours, switching gyms entirely - it's still the superior exercise. If not, the HSPU is a fantastic choice as long as you're aware of the points above.