Well, that didn't go well. I am morbidly curious if going to Taiga and Passionlip would go this badly as well. Hard to say. I've never seen EXTRA, but my understanding is that Passionlip is a little more passive, but prone to crushing those she "loves" into cubes and putting them into her extra-dimensional chest pocket.
[X] [Stall for Time]
I say we stall for time. Again, I don't really know much about EXTRA or BB from her original game, but it might be better to assume that BB just has total control and awareness of cyberspace, which means it's not impossible for her to be aware of Shirou's connecting to Caster, and be able to prevent him from doing anything. Or worse, just bodying Caster when she is summoned and torturing her. It would probably be better for Caster, or whatever theoretical third party, to do their thing away from BB.
And if BB doesn't like Shirou stalling? Well, Shirou is a glutton for pain, so it should be fine.
Seriously though, just going along with BB seems like a bad idea, and I don't see use being able to do anything to overpower or fight BB, so I'm going for stalling.