An immortal child pitching a fit. Honestly, that was what was going on here, and Carlisle kept calm as Claudia got out her tantrum. "Because there are other ways to survive in this life," he told her with a sort of serene calm that only Carlisle could pull off, because it was so much a part of him. He was good at this life, great at it, in fact, and though he could smell Bella's free-flowing blood, rich and heady, it did nothing save remind him that she'd have to be tended to, immediately, if she didn't want to end up bleeding out, or getting infected. "There are other ways, and yes, I'll stand protector for her," he assured the little vampire, his hold on Bella loose enough for her to get out of, if she wished, as apparently she did.
And that split his attention. "Bella, please," he called, his calm voice tinged with a bit of insistence. "Stay with me where you're safe." He had every reason to think that if he let Bella go on her own, not only was she at risk from her injuries, but there was a good chance that Claudia would find her again the moment his back was turned.
He didn't know Bella personally, of course, but that didn't stop Carlisle from wanting - nay, needing - to protect her. He could only thank God that Edward was sufficently distracted with Juniper, as the scent of Bella's blood did horrible things to the boy when it was safely inside her body. "She is the daughter of my friend, and therefore my responsibility," he told Claudia. "Please, Bella, for Charlie's sake, come back here," he begged, still not taking his eyes off the vampire in front of him.