Hat STYLE. Not contents.
That's for when you're a publisher representing multiple artists who are affiliated with the different PROs"You can use the same business entity (e.g. LLC) when registering as a publisher with ASCAP and BMI, but your affiliation name should be slightly different with each society. Acme Publishing LLC could become Acme A Publishing LLC and Acme B Publishing LLC, for instance. One extra letter is all you need to differentiate them.
For more information on personal publishing entities, check out our blog post here."
I'm looking at my (songwriter) agreement with BMI right now, and it says that all my songs - or shares of songs - have to be registered through them, and that no other PRO is to have claim on them. If they even suspect that ASCAP is paying me for one of my songs, they reserve the right to withhold payment until I prove they're not.
edit: I realized that saying all my songs have to be BMI and not ascap is misleading. I can split songwriting across the services if I wrote with an ASCAP-signed artist. So clarification there.