It was clear from the very beginning that the Crowned White Walker never planned to spare Kitsune.
The moment he saw Max, he had already decided what to do with him.
Devour him in darkness.
The desire to do that caused him to pursue Max without pause.
Unfortunately Kitsune wasn't stupid enough to be deceived by his words because of two things actually; she could sense the truth of all things, including whatever the White Walker said.
Even without the Eye of Munin, using Emotion control would allow her to understand his deepest thoughts.
Thus from the very beginning, she only wanted to know if he would retreat the moment he realized that they knew what he was after.
Unfortunately, that didn't seem to be the case as he prepared fight.
Kitsune clicked her tongue at the annoyance that was the White Walker.
She was down to fight but the fact that an army could be on its way over made her feel the need to quickly reach the Moon Palace go discover the truth about the Land of Snow.