I think some people are misunderstanding here, the problem is not that King is ignoring Kyle's time as a White Lantern since the beginning I figured that Kyle sees himself as a Green Lantern in his mind (which doesn't make much sense since he has gotten so many and more important identities over the years that being a GL shouldn't be the most important to him logically speaking but whatever).
The problem is that most of his character development is apparently being ignored. Kyle is not someone so insecure that starts doubting his whole value as a hero and as a person just because of making a mistake (at least not at this point). Hell, during Justin Jordan's run on New Guardians he made Kyle face several challenges that could be considered way more dangerous and complex than this one and he always found a solution instead of simply crying in a corner. Kyle is an artist, let him be creative.
Another thing that is kind of a nitpick, Kyle is a romantic not a cheater. He's currently in a relationship with Carol so him getting on with the princess is also out of character.
That's not really true though. Lemire acknowledged EVERYTHING that happened before his run, even Nocenti's run. There were several mentions during his work that Ollie was too busy traveling around the world (which happened during Nocenti's era) and there's also the time when the leader of the Shield Clan, Kodyak, say that he was called to fight in the Outsiders War just when he was in Amsterdam having a thing with some Danish twins, that's a direct reference to Nocenti's first story about the Skylark triplets and how one of the sisters died at the end.