I don't see why you want to persist with the Q2/Q3 arrangement that allows shoot-through. Sure you've got the resistors there to limit the shoot-through, but why not just use the class-B (common emitter push-pull) stage as previously suggested. Then you only need the one current limiting gate drive resistor.
If you really want to do it with discrete components and want it as simple as possible I would have suggested something like the following.
BTW. You can increase that 15 ohm resistor quite a bit at the relative low switching speeds you're using. You may need to go up to about 150 ohms if you want to keep zener power under control now that you want to run it at 22 volts.
