It'll ask you to pay for the battlepass. Even asks you if you already have the battlepass. Kind of annoying. Also asks for money for BFPro, even if you already have it. You can tell what they really care about with every login.
Anyway... No, you do not need to buy anything once you already have the game.
If you want to get a battlepass... Don't do it right now. The new season starts next week so the current pass will expire and be over. Money down the toilet. It'll be a massive time crunch to try to finish it in a week. You'll get the benefit of having everything available to unlock all at once, but I worry that will be impossible to finish anyway, especially in a week. I finished the Season 1 pass in around 220 hours
Play for a week, if you like the game and want to continue, I highly recommend getting a battlepass once next season starts. You'll get gun unlocks early that really help.