Not sure what the trigger is, but I did not have Alfred summons available for BSB on either NG or NG+. I did VA before BSB each time. I know I talked to Alfred and told him I would cooperate with him each time prior to getting to where his summon sign is for BSB. But I also opened password door (but didn't go through) before going to BSB, so may that is the trigger why I could not summon him.
I am not sure what the trigger is whether he is summonable or not, but it must be something I missed on ng or ng+
I never summoned him. wasn't sure how either during NG. I didn't even bother going back to BSB in NG+. heard he basically destroys BSB.