I'd propose that Kore is just a Paladin of a god who is a shitter, and believes in the sham of moral relativism. Basically Kore's god says:
1: I support my worshippers.
2: My worshippers' needs matter more than the needs of other creatures.
3: My worshippers consider that their interests and survival are Good. Things that work directly against their interests and survival are Evil.
4: What my worshippers consider Good and Evil are what I consider Good and Evil.
5: My Paladins must be Good.
Conclusion: My Paladins must do whatever aids my worshippers and must fight against whatever hinders them.
So Kore can dial-a-Commune and his god says, "Hey mate, goblin genocide alright?" and Kore is all like "Sure thing boss." If Kore ever hinders his god's worshippers that could be an alignment violation, and if he fails to take advantage of ways he could aid them that could be an alignment violation. But anything he does that's not covered by those things isn't considered in his religion's morality, which means other actions don't affect his alignment.
Tough decisions involving hurting some of his people in order to help others of his people are gonna cause Kore turmoil. That's why Paladins make poor politicians.
Now consider further that if Kore's people ever die out, that is, not all Dwarves but just the worshippers of Kore's god, then Kore could be on a never-ending quest for vengeance. The potential vengeance of Paladins acts as a deterrent to those who would attack the worshippers; fulfilling that vengeance is similarly part of his moral duty. Any creature who ever hindered Kore's people would be a potential target for Kore's genocidal vengeance.
As I said, I'm not really arguing anything here, I'm just putting out there that Kore and his god could be using the fallacy of moral relativism to let him keep his Paladin status despite his shittiness.
Or he's a homebrew non-LG Paladin. Whatever.