1) I wouldn't add a shoulder/ohp session if you are already bench pressing often. It might do more harm than good. Rather it would be smarter to add some overhead work on your bench press sessions.
2) The push press isn't really more of a triceps movement. Sure you are handling more weight, but that's because you are creating momentum with the leg drive, so yeah you are locking out more weight, but the momentum does a good job at not increasing the demand on the triceps. If you want to work on triceps strength via overhead pressing a better option is top half shoulder press from pins with the bar starting about an inch above your head. The push press is a good exercise but don't see it as a way to boost triceps strength.
3) Another good movement to increase triceps strength via overhead pressing is military press with added bands.
4) If you are far away (8-12 weeks) from your competition I would put one overhead movement prior to your bench press. When you are 6 weeks out put it second after the bench press, at 4 weeks out do it last in your session, at 2 weeks out, drop it from the program in favor of more flat pressing.