The evidence that King Arthur did not exist would be the fact that there is no evidence of his existence (according to your post). You can't look for evidence that something did not exist, you can only conclude such.

What would count as evidence that he didn't exist?? It's a weird question, the burden of proof is upon those who claim he did exist, not those that say he didn't. The "evidence" of him not existing is the lack of evidence for his existence.

The problem you have right from the off is defining who Arthur was. It's similar to the debates on Jesus' existence. Yes, there were probably lots of people called Jesus in the region at the time. Some probably even did some the things the Bible said The Jesus did. But the issue is deciding which attributed stories and traits you accept as belonging to the real figure and those that you don't. Arthur, as we know him today, is a patchwork of various british and french writings over hundreds of years. You need to define your Arthur first and as he's pretty much only present in fictional works, it's pretty much impossible to say whether a similar figure existed in history or not. Is the Arthur you're looking for Monmouth's Arthur? or Malory's Arthur? Or Chrétien de Troyes' Arthur?