Once you are streaming here, enable a feature in your account manager named "Enforce Bans" and disable "Accept Bans".
How this works:
Any player banned at PsB will be unable to join your server, since you will have used their Auto-MBL to update your server's ban list.
Any player who is not banned by PsB, will be able to connect.
When a player connects to your server successfully (i.e. not banned by PsB), the PBBans HUB will check their GUID against the PBBans ban list. If they are banned here, the PBBans HUB will kick the player from your server and add the ban to the ban list.
You will never need to download a ban list from PBBans, or update it. The HUB takes care of everything for you.