Yes you would need your co-writer’s PRO and CAE/IPI#.
If the song is self-published and either of you have publishing companies, you would need those company names and CAE/IPI#s as well.
BMI allows you to collect the publisher’s share as “excess clearance” without having a publishing company. I assume your co-writer is PRS. I don’t know if that’s true with them or not. If we have any PRS members here, they can chime in.
You should be able to contact BMI for assistance on how to fill out the online registration.
Good luck!