Exorcism (For those sticking around after the rescue)
The rescue had gone off without a hitch. That was good. Buffy hadn't really looked anything near good when they'd found her, and Pike's guilt had magnified by about a thousand. He'd promised himself when he'd seen her strung up on that wall, barely conscious and in varying states of injured, that the next time she told him someone - anyone - was acting off, he'd make with the holy water splashing immediately.
After Buffy had been freed, Pike quickly hashed out his plan for the exorcism with everyone. He wanted the extraction team to know it in case things got hairy and some of them needed to be recalled, but all they needed was the basic gist. He didn't mention that he knew he would probably need a doctor after this. That would be evident to anyone with eyes when it was all over, anyway.
In the end, the exorcism team was himself, Andrea, Spike, Castiel, and Sam. On the whole, Pike was pretty pleased with this. He trusted Andrea and Sam, and as vampires went Spike was actually pretty okay. And Castiel had been helpful so far, so Pike was willing to extend his trust a little farther. Once the extraction team had left, Pike quickly hashed out the more detailed version of his plan with the rest of the team.
It was pretty simple, as plans went. They would hide until the demon came in. He'd play lovesick-lone-gunslinger and keep it distracted while they kicked into gear and got a trap drawn up. He hadn't told the extraction team the part about the human bait, or why specifically a doctor was going to be necessary. Wouldn't be hard for whichever doctor showed up to tell, if he got even half as trashed as he was figuring he was going to. After the trap was drawn, they'd get the demon in there and make with the exorcism.
Until then, there would be a lot of waiting. At least, that's what Pike thought, but the demon arrived sooner than he'd expected. Oh well. The others were already hidden. He hadn't had as much time to come up with some stupid lovesick crap script in his head, but that was okay, he could wing it. "Lucy, you got some 'splainin' to do," he joked, wagging a finger at her comically.
His sarcasm-rich mirth faded. Cue the lovesick puppy routine. "Faith, I know you're in there. I know you can still hear me. I'm gonna get that thing out of you, baby, you hear me? It's gonna be okay now." His face hardened and he pulled his hand from his jacket, popping the cap on the flask of holy water. He immediately started the Latin, knowing the demon could and would stop him. That was the point. Keep her focused on him so the others could get the trap drawn.
Once the beating commenced, he'd put up a token fight, throw in some cliche pleading lines about knowing Faith was strong enough to take control back, and when it got really bad throw in a few 'it's okay, i'll always love you no matter what' lines. Catch was, it was all technically the truth, but he knew possession didn't really work like that.