I definitely feel like the chanting is "Cintamani" instead of 'chains of harmony,' and "eminent form" might be 'tormented form'? I don't feel like "An empty heart divides the sea" is right either--but I have no alternative to offer. I'm also not a fan of "One must start believing;" but 'One must' is definitely true, I think.
Like Mysteltain, the mother-daughter dynamic is most interesting, doubly so we seem to have two deiforms from but one Primal. I'm going to piggy-back on the current trend of theories with the English eikon being a deified object as opposed to idealized things here, but it's necessary to draw a key distinction.
Cintamani, the namesake of one of Sophia's attacks, is apparently a wish-fulfilling jewel much like an Eastern analogue of the philosopher's stone, if my thirty seconds on wikipedia is even half right. The first few lines paint us an image that almost immediately turns dark, I feel. A red warm dress screams bloodied to me, so there is a wounded woman in a dress; given mothers and daughters are typically women--not that men can't wear dresses. Her memory fades, a tormented form, desperate pleas, farewells unheard-- one of them is dying. Rather, one of the pair is dead in a silken dress before proper goodbyes are said. Giving us Barbelo as the name of a "mother" figure is helpful as well, but we're then lead to believe the head beneath the heels is actually the daughter (Sophia speaks to her 'daughter' and Barbelo disappears to perform an attack).
Forsaken rites, prayers, a whisper and then a Goddess speaks to a soul--quote a poetic description of Tempering. Like her Meracydian brother Bahamut, could Sophia be the ideal of a lost loved one? This is... complicated and probably wrong, but bear with me. Bahamut, Shiva, et al. are the ideal of the person--Tiamat herself knows the primal to not be her brother on sight twisted by her rage as he was and Hraesvelgr chastises Ysale for being so foolish as to believe she had brought back his Elezen lover. So then, what makes Sophia an Eikon?
Well, Barbelo is a floating head. Where's her body? Yes, yes, I know "Nomura didn't draw it!" Could it not be as simple as a decapitation? That would certainly stain a dress red and torment a form, for sure. Somewhat comically, when Barbelo attacks, she rears back and there's this sort of appendage that flits about I think is quite like a tongue. I don't think this strengthens my case any, but I couldn't resist mentioning.
Zantetsuken, the deification of a sword. Sephirot, the deification of a tree. Sophia, the deification of a loved one's head. Please god don't tie this to Edda, but holy hells if that misguided relationship isn't at least somewhat similar. This distinction would draw her eikon and not Primal, as this forsaken ancient rite did not literally produce the idea of the woman she wanted but perhaps she wanted something as simple as the eyes to open one last time.
This thread of revivification ties in well with Cintamani and the actual fact the "zombie" status exists in the fight. What does she say when someone dies during the Demiurges again... "Death will offer no release"? I'll have to pay more attention next time.
The final and most damning evidence I have for my own theory is has precious little to do with balance or equilibrium and the like. There's also the fact this would be the song about a summoner rather than the devoted thralls. "Sin must be balanced with suffering," hearkens back to a very archaic idea of justice for me, and so I posit the Goddess is seeking to exact justice in some way for the murder of her beloved. Mayhap her beloved was killed as a consequence of some archaic justice? Decapitation as acapital punishment is certainly not unheard of.
What most of this lacks is Allag though. Did Allag kill the beloved woman? Or perhaps, more likely I feel, we are placing too much on the idea that Sephirot and Sophia are "new" gods in the Third Astral Era. It's possible I'm forgetting some key detail somewhere, but could it not be that these gods are not new, but simply the first summonings? Faith in the Twelve has survived at least four Astral Eras. What we know of the Sixth Astral Era, there were some thousand odd years before the first rumblings of summonings amongst the Beast Tribes, but surely they still had these Gods? Mayhap the mother-daughter initial summoning and eikonification of severed head happened some years past and over the ages the wrongness of justice mutated into the balance we find in Meracydia's Sophia-- but the actual execution is removed from their Allag problem. So, we have a beloved family member crying out over the twisted remains of their kin crying out with a stone and prayers and something answers her; bringing together he beloved family as one being and folding over herself. The aetherial shade which answered Sophia is the Goddess, and the 3 consciousness intermingling gives us what we see.