It would have been a highly unusual move for him to be Arthur II, as it would have broken the previously established numbering system of English kings. This is entirely independent of whether King Arthur was considered a real historical figure or not. The root of this issue lies with the three Edwards. Edward I of England is not the first English king to be named Edward, not by a long shot. In fact, he was named for Edward the Confessor (1042-1066), who was not himself the first king Edward. There was also Edward the Martyr (975-978) and Edward the Elder, King of Wessex (899-924). This latter monarch may or may not have counted, as Wessex -> England is complicated. Still, Edward I was by any reasonable reckoning either the 3rd or the 4th King of England named Edward, but he is known as Edward I.

I've written about this previously, but essentially Edward I is Edward I because he was followed by Edward II and Edward III. In his life, Edward was simply 'King Edward' (or 'Prince Edward'). His son would have been 'King Edward' or 'King Edward, son of King Edward', or possibly 'King Edward the Younger'. All were reasonable ways of distinguishing individuals in the Middle Ages. When Edward III came along, though, it seemed to be too much and it is around that time that we begin to see the three Edwards with numbers. Early on this would have been in the style of 'Edward the first since the conquest', or similar, making a clear distinction between monarchs pre-1066, and those after. This style remained for some time, and was even used on numbers for monarchs who had no pre-conquest equivalents (the Henry's for example), but was eventually dropped from official use, leaving us with just Edward I, Edward II, and so forth.

As to why it was the Edwards who caused this situation, they were the first time that a post-conquest king had the same name as a pre-conquest king, while also the first time that any post-conquest English monarchs had the same name three times in a row (also maybe pre-conquest, but I'm shaky on all things pre-1066 so I'm not willing to commit to that).

So, essentially, to give Arthur the title Arthur II would have been to break the established numbering trend for English monarchs, which clearly began with the Norman conquest. That is not to say they couldn't have done it, it would hardly be the only time the Tudors broke with tradition (reigning queens!? What!?), but it would not have been the norm, regardless of the historical realness of Arthur.

Marc Morris has some discussion of the Edward naming situation in his A Great and Terrible King, but it's not super in depth. A lot of my recent information came via an amazing lecture given by Robert Bartlett. I believe he intends to publish his talk in an edited volume (or at least he hinted as much at the time), but since the talk was only earlier this year, I wouldn't expect to see it in print soon. :(