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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.

Oh dang, you're right. It seems I've got myself confused on the antecedent of that 'he', hehe. Looking at it closer and in that context, it does make sense. Those smells I believe are still distinctly Asterion's though. Seawater and crocuses were referenced a lot in Asterion's history; I've seen this throughout his ruminations and the clay tablets. I'd chalked off seawater to Poseidon, and crocuses to Crete with their numerous depictions of the flower and its saffron. Maybe he rubbed off on us while wrestling? Or, as I've come to believe (thanks to your reply) is a reenactment of Asterion's fight/execution and funeral.

Let's sort it out detail by detail:

• While there, the Master and Asterion make a fire. This parallels Asterion's funeral; his body was put to rest by way of a pyre, burning him on the very basin meant for Hestia's flame. Additionally, both are forms of divine worship. The fire reminiscent of his duties to Hestia  in the original labyrinth, and his death was construed as such in the underworld trial. Getting back on topic, the fire made with the Master does something strange. Earlier in the scene, we get this:

The minotaur and you summon meat and fruit, wine and olive oil, dittany and crocus-- all of it fed to the dwindling fire until it shines proud once more. No foul odor wafts from it, neither do violent embers jump out with your offering.

This follows a bit later:

The wind blows. A burst of embers dance about in the nude breeze, catching your attention. The air around you is filled with the smell of fire, smoke, and burning wood.

The wind and fire acts up just after Asterion plans to get rowdy with you, asserting himself with renewed confidence.  This is probably the start of weird things happening here. Birthmark alert! Yep, definitely important.

• Asterion's scent suffused the whole scene as he was a major participant in his execution. Specifically, the smell of seawater and crocus flowers can be attributed to his divine and Cretan origins.
• Deflowered poppies are also smelled. Poppies are associated with death, which is of course relevant. The addition of the 'deflowered' adjective is strange though, oddly specific. A deflowered plant has had its flower picked-- suggesting the use of poppy flowers for something, like an offering. Maybe Laomedon included some poppy flowers in Asterion's funeral (No such detail in 17th tablet, sadly)? Of course, this may just be for literary effect-- deflowered/picked flowers are subjected to death, after all.
• Burnt oil and coppery taste was noted. Going back briefly,  Asterion was put to rest by a pyre. Burnt oil may be ichor being burned; his body and the basin where his pyre was was filled by such fluids, after all (17th tablet). Coppery taste is probably (mortal) blood, then.  
• The Master's smell follows after this. This signifies the presence of the redeemer that has come for Asterion. It came in the form of Theseus in the past, and now, you take his role.
• Asterion experiences a searing pain around his neck, reminiscent of his beheading. Theseus's coup de grace imparts a lingering scar on Asterion which flares up at this moment. And you, the master, deliver your own brand of mercy by finally ridding him of this.

 Asterion laughs, and yet his throat constricts to choke him halfway through. A ring of fire burns around his neck. An unfamiliar pressure wells inside his chest-- expanding outward, like a star about to burst, until it becomes unbearable. He opens his mouth to cry out, and that's when he feels a hand brushing against the burning collar which threatens to strangle him. A gentle, cold touch, bringing with it liberation like no other. With that the ring of fire breaks for good, and the overwhelming pressure within him can finally burst out.

Yeah, you love this part too. Damn writers making us feel good.

• The wind picks up again, accompanied by the telltale shuddering of the labyrinth and its obsidian in particular. Act finished, draw curtains.

Yep, really digging where this line of inquiry stumbled upon, hehe. The achievement's name and description has never looked more apt with the reveal of more and more of those details. As for a ritual taking part, I'm not too sure. It has a lot of similarities, but differs in the common component of the rituals (we've seen so far)-- the creation of a ritual focus (e.g. Storm's ear gauge, Pedro's bracelet). Who knows, though? We don't know what the other rituals are, and Pedro's are only a subset of them, I imagine (with the authors' push on different kinds of magic and their authenticity all throughout the world and cultures). With that said, something really significant did happen, with the hotel getting the jiggles and all. 

Yeah, I think you're spot on. I'm going to shamelessly repost what I wrote in another thread that's focused on the tablets, because I hadn't connected a lot of specifics that you've found. 

I've been waiting to read the tablets until I collected all of them. Having done so, I think it does indeed point to his thread being restored. 

This is all my interpretation based on the tablets and the ruthless route, but it seems Asterion was given two options to escape. The first would be to take freedom for himself given to him by those who pitied him or regretted their actions. The second would be to right the wrong he was accused of by actually fighting an opponent who wanted to defeat him. The result wouldn't matter, but he absolutely could not back down. However, he would need enough freedom from a master in order to fulfill this, which is why he's never been able to until now. He needed to stay a prisoner of a labyrinth in order to gain freedom and have his ember rekindled from nothing, essentially, and the rules were created to stop that from happening.

I think Poseidon's gift, which relates to what he sees as a moment of weakness when he questions whether he should escape by sea, is a backdoor. A failsafe. I also think the master situation is a cruel repurposing of his refusal to leave without permission from Laomedon, along with his eventual crime.

Your analysis of the specific meanings is great and it's making a lot of things click that I hadn't caught on to.