I thought this question might be interesting due the continued interest with chivalry into the Tudor age and the ramping up of the Tudor propaganda machine (which perhaps may have liked to play on chivalric mythologies and romances). Even with the continuity of Regnal numbers starting at the Norman Conquest. would Arthur II have ever been considered?
A follow-up question of sorts:
Did other monarchies, Duchies etc. use legendary people as part of the numbering for names?
Like, suppose the Duchy of Toyota had a legendary founder Duke Camry, did the first actual Duke Camry take the name "Camry II"
Would he have been known as King Arthur at all or for tradition's sake would he have been assigned a regnal name?