A poly relationship likely isn't in the cards, there's only been one thing I could point to that compared to the boundless other examples against it, but that's the only way I see a romance working. Not just because Asterion needs trust, so a scene like what you're describing would be devastating, but also because Argos genuinely wants those two to be together. I do see something between the three (depending on some hyper-specific choices to see what I mean), which is close to what I'd consider an ideal relationship where it's 3 people with equal feelings, but at the end of the day this is a story about breaking cycles and I think it runs too close to repeating Laomedon's split interests if any other love interests, poly or otherwise, are added.
(I would obviously be happy to be wrong, though! I just think it's harder for a story as seeped in symbolism as this to pull off in a way where the significance isn't muddled.)