It comes down to the simple fact that you are an outsider. You are always going to be an outsider, no matter what you do for the den or what you become. You will always be an outsider and never one of them.
I would imagine that a great deal of resentment comes from the fact that Champion can and does do much that reminds the Kitsune of their collective shame;
- The Champion can defeat several individuals who defend the den, only stopping when they meet Komari and show the amulet. The key home point for the Kitsune and this barbarian (classical definition) kicks the door in and kicks the ass of several people, including the "Commander of the Western Colonial Garrison", which kinda shows how poorly the Kitsune might defend themselves from an actually aggressive attack.
- Helps resurrect one of the scions of one of the ruling families in the Old Country, but also helps produce a child marked for greatness.
- Becomes quite friendly with Keros, a rare honour it seems, especially for a colony that many in the Old Country would probably wish they could forget about.
It always seems as though the Champion repeatedly shows up the kitsune, either reminding them of their clan's shame or perhaps how inflexible they are. And they probably know that, and they probably resent that.
The kitsune who actually like the Champion, such as Kohaku and Kurako, probably, in Kurako's case, the Champion is fun. In Kohaku's case, it probably goes a little deeper. She's a merchant, she's not from the clan herself. She's as much as an outsider as the Champion, if not more so, due to being a merchant. A necessity for the survival of the den, but in the same way that fertilizer is necessary for the survival of crops. Doesn't mean that people want to be putting their hands into shit all day, pardon the crassness. Whereas in the Frost Marches, to be a trader or a merchant is seen as an honourable career, so for Kohaku, to have someone who treats her like another person is probably a rare job in the den and explains why she often spends time travelling the Frost Marches.