Do you get battlepoint when leaving a match early?
Solution 1:
Upon death in a team game, your points aren't shown to you until the rest of your team is either killed off or wins. I played some duo games with my friend and at one point I was killed, and it didn't show them. My teammate was still in the game at that point I backed out to the lobby. Again, I tracked my BP, and it showed that after quitting to lobby, I was given points (I don't know how much, but it appears I only got position points since I didn't shoot or kill anyone).
In a Solo match I tested, I landed fast and got a kill quickly and exited to lobby. Prior to the game, my BP total was 9,656. After getting my kill and exiting to the lobby, it was 9,726. I got 40 points for the kill (20 for damage, 20 for the kill) and apparently an additional 30 points (I assume for the position that I made it to). I see no reason why this wouldn't work the same way for a team game mode. This leads me to believe that your BP is granted immediately upon an action being performed (position change, damaging someone, and getting a kill).