It's definitely a team effort here, I wouldn't've started looking down this path without what you've already posited!
Interesting idea about having Asterion be able to add his own addition to the constitution, I think a fully divine Asterion would be able to do that, yes. If we follow the theory that Asterion's birthmark coming back is a sign of his rekindling divinity because he believes in himself, I think he can build up to have this effect, and it explains why even if he understands the obsidian and bedrock, he has never done it before because it wouldn't work for his 'blasphemous blood'.
Speaking of said blasphemous blood, there may be a loophole here in this 2nd quote if he fully ascends to divinity somehow: pure divine ichor clearly can't be blasphemous, right? I'm not sure how we break where it calls out his name, but at least here's an idea that can handle the latter half of that sentence that would prevent him from leaving.
Mmm, good point on how it specifically says rhyton. With what we know right now, the threadcutter is either Labrys, which we have the axe head of in storage (Where's the rest? What's it shaped like?), or Asterion himself, yes... maybe in a less morbid way it's the rest of Labrys or just one of his horns snapped off and since regrown? googles 'bull rhyton' Yikes, or maybe not.
Still, whatever the rhyton is, I'd agree that if Asterion can muster up more ichor than human blood and add it to the rhyton, he should be able to affect the realm in some way, and possibly amend the constitution at minimum.
EDIT: Oh forgot the point you had on that one ruthless ending. I think it's just further reinforcing the theme of recursion, and it's also highlighting that the pelt still counts as Asterion in some ways for the realm's constitution. That's also the only ending where there's someone around for the pelt to graft on to with no Asterion around, so it might not be tied to Asterion's subconscioud but instead the laws of the realm just in general... or like, whatever the eff is going on with pelts in this game, lol, the Tapir God's nearly did the same to P.