I assume that if a non-bishop held the Patrimonium then the name would change to something more appropriate? Some variant of Duchy of Rome perhaps? I feel like the holder of the papacy should, by default, get a strong claim on the Patrimonium if he loses it which can be inherited by his successors.And because, yes, there are situations in which the King of Patrimonium Sancti Petri would not be the Catholic Pope. E.g., he could be the Fraticelli Pope.