The answer to this is simple. We roll them militarily or at least threaten to do so, put into place a local government that is required to follow our human rights laws like anywhere else in the HR, and then backstop them with military force.
Question is how we want to handle this, in terms of details and resulting optics.
Right now I figure we'd contact the various lords of the equitorial nations and demand they hand over the demagogue priests, and the culprits who attacked villages and schools under our protection, over for investigation and trial.
If they decline, we have justification for military action under the guise of them housing and protecting terrorists who participated in an attack on people under our protection. If they agree, it kicks the can somewhat down the road, but they'd weaken their own authority and legitimacy in doing so, which could easily lead to unrest as vassals and religious leader rebel against them due to a combination of ambition (the lords being seen as weak due to submitting to outsider demands), and fear/concern (that they, too, could be handed over if a similar situation occurs again). This would then likely give us/our "vassals" on the planet reason to intervene in order to restore order.
This would be an approach that makes us look the most reasonable/sensible, but runs the risk of the NO situation remaining as it is for quite some time longer.
More straightforwardly, we could simply send in some troops - likely in the form of military police - to apprehend the demagogues and culprits for trial; still fairly reasonable, but much more heavy-handed, and quite possible to escalate into a full-out military confrontation.
We could also potentially pick up your old idea of simply bribing the NOslo nobles, but with a bit of tweaks to account for the fact that we now basically have their balls in a vice, since we now have a fairly easily justifiable casus belli for our own population (because I wouldn't be the slightest bit surprised if those attacks and burnt-down schools didn't involve at least a couple of dead children, which our populace is unlikely to be all that happy about).
So, on one hand, the stick in the form of the threat of military intervention that'll almost certainly see them dead or deposed, and on the other hand, carrot in the form of a comfortable retirement somewhere nice, with some possible face-saving measures such as them abdicating in favor of a son who'll be happy to work with us, or them marrying a daughter of theirs to one of the polar lords who're part of our protectorate and who'll absorb their lands.
A) Right now, the primary industry of New Oslo is probably subsistence farming. We have breadbasket worlds that are about to have a surprise once Helghan terraforming progresses. We can make agriculture no longer an industry that requires most of the population.
B) As a result, we can have a large number of people in need of work. We can fill that need by literally building new cities en masse and putting in subsidized worker cooperatives to produce things for themselves to allow the planet to work towards self sufficiency. What we can then do is carefully stagger what jobs are offered where and to whom to mix the population. It's also a lot easier to pop a panopticon and prevent rebels with Explosives or firearms from causing atrocities.
C) We need to identify and support moderate religious figures. They don't need to be as leftist as the HR is, they just need to not be utterly shitty people. If we don't do this, the people who continue to preach will be the radicals and they'll do it under wraps.
D) Education and medicine is critical. Education shifts people away from conservative badness. Control of fertility gives women power. Both of those things will permit us the break the stranglehold the nutcase have on New Oslo.
We'll probably do A & B as par of the general "technological and economical uplift" thing that we're doing on most of our worlds pretty much by default.
C & D are both good ideas/points, though in regards to D we IIRC just had another case of "you can't expand your education system that quickly!" on Highwater, so that might raise its head again.
I'd add that we should make efforts of cultural and personal exposure; make it easy for NOslo inhabitants to get access to various ORDI media (and the necessary technology to use them, of course), such as movies, books, games, and the like, many of which would feature girls and women as protagonists, in positions of power and authority, and so on. And make sure that a fair number of personnel we send to NOslo includes women in various functions and levels of authority.
