Could it be that Fitzwilliam is a reference to Arthur Grey himself. Fitz is a patronymic prefix used in many Irish names of late medieval origin, signifying "son" (like Mc, Mac, O', and the Hebrew Ben). Emmett O'Byrne's excellent essay mentions that Grey's father was William Grey, making Arthur = Fitz William. Put in context of the lyrics it makes sense, especially since he was the Queen's representative - what better than to send his head to her.