He's the strongest person in the land or so it seems, so there's not much the kingdom can actually do against him if he up and disappeared. My only hang up with that is the son, but it also looks like he doesn't have a great relationship with the child like the mother does so at this point, yeah I'd say it's best to.just leave and maybe drop a note as to why or packnhis bags, tell the king and roll out with the Warrior woman.
Yes, that's likely to cause some political turmoil but at this point the royals, at least the Princess is just using him. I think it said that the King saw him as a son. I also think that this is partially the MC's fault for likely not saying anything about this until now. Still, I don't see him staying with the Princess unless having concubines is a thing in this world but then that would likely put the Princess in a bad light because MC is likely to develop a better bond with the Warrior lady. So despite the Princess' lack of interest in sex, she'd definitely get jealous of the attention he gives the other woman. And then there's the Princess' words which implies that she believes that the MC shouldn't be having any sexual urges at all which challenges the idea of concubines in a way, at least for this kingdom.
So that leads me back to he should roll out as diplomatically as possible. It will cause some issues but that's what I'd do. Either that orvstay ti raise his son to an adult and then roll out all the while distancing himself from being a king candidate. We already know he's not fully invested in that anyway.