It does take a moment for James to register that 'Prior' is the other man's name, so although the added clarification isn't needed in itself--it's an assumption he never would have made, no offense, Prior--it does actually help. And he appreciates the sense of humor inherent in it, as well as the comment about Last Rites.

"Fitzjames." He introduces himself in return, allowing Prior to adjust the coat without protest despite his disinterest in being fussed over. "And fortunately for us both, I am very difficult to kill."

Even when he wants--

He's not going down that line of thought and instead redirects back toward Prior, having not missed the pauses between his words or the sound of his breathing.

"Is there something that can be done, for you?" He doesn't know what helps, but if there's something he can do, he wants to.