I think that general theme you should stick to here is that Kord does not 'grant' jack to his followers. His followers take what they are strong enough to take.
My suggestion is:
1. There is a well renowned and honored Kord worshiping barbarian clan. Say a bunch of chosen half-orcs. They are like Cimmerians from Conan.
2. The renowned and honorable leader of them is commonly regarded as an Exarch of Kord.
3. The players have some reason to seek this guy out, maybe as a mentor. Or to ask for their help in a worthy cause.
4. Before they arrive though, he is overthrown in ritual combat by a stronger guy (The original guy was quiet old). This guy is like Kurgan from Highlander. This guy has no honor, just strength. He leads the clan to break its ancient treaties and start raiding all the weaker clans and kingdoms (which is pretty much everybody).
5. The bad guy is now an Exarch of Kord, having taken the title and power from the last guy. Kord could have overruled that of course, but it is not really his style to do so. He will leave any vengeance to the next guy worthy to be his champion.
6. So now the PCs have to battle their way through this mighty and blessed clan, to reach the leader, where the warrior challenges him for the right to bear the Mark of Kord.
7. Kord is watching the battle of course, but is not really the sort to fawn over anyone. He sends a beautiful Valkyrie to bring a mug of ambrosia beer to the victor and "help him celebrate".
Just my suggestion, but I think it ticks all the necessary boxes.