And why would he have any responsibility to them that he might somehow "betray"? He has responsibilities to Family, to Love, and to his Court. He can be their enemy without betraying them.
Also, fwiw, the White Court as a fundamentally Mortal (IE still have Souls and Free Will) group of "half-born" wouldnt be subject to Oblivion.
The white court are probably closer to Outsiders then even the Ramps, the ramps demon comes through and finds a place here. Wamps straddle the line with their demon stuck on the other side, if how Thomas's demon looks in soul gaze is any indication. That's the part that makes them half born, yea? Of course that takes as a given that Oblivion is just the final step of Outside.
And I'm pretty sure anything that has any 'NN' blood or whatever you'd like to call is subject to also being forgotten even if unlikely because the main proponents are mortal themselves, case in point, wizards.
Heck though, lets go one level deeper.(can't remember which is which so 'Vlad' refers to daddy, Dracula to son) Vlad 'son of the dragon/son of the devil'(Lucifer=snake=dragon without wings=fallenangel.) has a son. Now we know angels have a one choice switch, either plus or minus. I think when Drac was trying to impress his father he accidently hit a switch on his 'big jumbo jet' that he didn't know either, what it was, or what it would do. He let the creature in the mirror through, became it even. Became the opposite mirage of himself, he fell.
So the black court and the white are connected by both being part divine/angelic(/outsider?) and the black court would have more reason to call Thomas a traitor because they still see themselves as part of the same stuff.
*kid screamed in my ear the whole time I wrote that, so if it seems choppy, it was.