Descriptions of House Rogarvia report them to be ruthless kings that ruled and maintained unity by intimidation and rule of authority. I'm assuming that Hellknights are exclusive to Cheliax and its vassal nations but I just keep imagining that the Rogarvias would have assembled a Varingian Guard-style company of Hellknights, seeking ever more effective tools of terror to compensate for not having dragons. It could also play to the rumor that Choral the Conqueror made a bargain with a devil for his rule by pledging his descendant's blood.
Can Hellknight chapters exist outside of Cheliaxian authority? Are exile, mercenary or rogue Hellknights at least a thing?
They can, although this mainly seems to be in places that used to be under Cheliax (such as Korvosa and Sargava); however, Hellknights have also been part of the Mendevian Crusades, and I could have sworn that I read somewhere that they even have an outpost in or near Magnimar (although now I can't find the references for these last two), and one of the Varian Jeggare Pathfinder Web Tales had Varian Jeggare having to put up with an especially obnoxious Hellknight on an expedition through the Mwangi Expanse.
I'm sure they'd be happy to let you help them to set up an outpost just south of Brevoy in Kingmaker.
And Order of the Scar is a mercenary (although sanctioned, with specific purpose) entire order, while Order of the Crux is an entire rogue (and now all Undead) order.
Given that the Order pf the Nail is very opposed to bandits and wild monsters (two things you find a lot of in Kingmaker), some of them could show up to help establish the rule of law over the Stolen Lands.