It's not mobility if you can lockout your elbows in a push press, it's just bad technique. If you fix the problems i described above, you'll be able to get the bar higher and get your body under it better, which will allow you to lock it out properly. If you and the bar are moving forward, you will never get a proper lockout.
Also try to keep your grip loose. Don't squeeze the bar like you would with a slow strength movement - rely on the legs to drive the bar up, then punch the elbows into full lockout as you split, making sure to allow your hips to move down far enough. Think of trying to lock the elbows at the same time your feet reconnect with the platform.