While you could probably get a healthy PC for competitive gaming for your listed budget, I'm not convinced it's the best route for you to go down. I'll put a spec list together for you in a few assuming that nobody else does, but is there no way to disable cross platform for those games?
People running PC setups might well have advantages for a multitude of reasons, but that doesn't mean you'll be better by switching and in some cases there's people who simply cope better with a console set up.