Let's start with the answer you mentioned:
Did Anakin truly fall?
Absolutely, and to such an extent that he couldn't be fully redeemed, according to George Lucas. Remember, this man has killed dozens of children - both Jedi younglings and Sand People children. He was actively involved in the destruction of an entire planet. He has tortured and murdered too many people to count.
How was he redeemed?
Through committing self sacrifice out of love for someone else - love that was stronger than his love for himself and the Emperor.
"It really has to do with learning," Lucas says, "Children teach you compassion. They teach you to love unconditionally. Anakin can't be redeemed for all the pain and suffering he's caused. He doesn't right the wrongs, but he stops the horror. The end of the Saga is simply Anakin saying, I care about this person, regardless of what it means to me. I will throw away everything that I have, everything that I've grown to love - primarily the Emperor - and throw away my life, to save this person. And I'm doing it because he has faith in me; he loves me despite all the horrible things I've done. I broke his mother's heart, but he still cares about me, and I can't let that die. Anakin is very different in the end. The thing of it is: The prophecy was right. Anakin was the chosen one, and he does bring balance to the Force. He takes the one ounce of good still left in him and destroys the Emperor out of compassion for his son."
- George Lucas, The Making Of Revenge Of The Sith; page 221
"You learn that Darth Vader isn’t this monster. He’s a pathetic individual who made a pact with the Devil and lost. And he’s trapped. He’s a sad, pathetic character, not a big evil monster. I mean, he’s a monster in that he’s turned to the Dark Side and he’s serving a bad master and he’s into power and he’s lost a lot of his humanity. In that way, he’s a monster, but beneath that, as Luke says in Return of the Jedi, early on, “I know there’s still good in you, I can sense it.” Only through the love of his children and the compassion of his children, who believe in him, even though he’s a monster, does he redeem himself."
- George Lucas, quoted in J. Windolf, “Star Wars: The Last Battle,” Vanity Fair, 2005
"And obviously there are two sides to the redeemer motif in the Star Wars films. Ultimately Vader is redeemed by his children.
- George Lucas
Now for your actual question:
Could Anakin have been redeemed on Mustafar?
Since he had already murdered dozens of children, among others, I don't think so. He had already become such an evil monster that what he did to Padmé was almost irrelevant. To redeem himself, he needed to sacrifice himself and everything he had.
Also, he wasn't redeemed simply because he loved someone. He was redeemed because he loved that person enough to sacrifice himself. His love of Padmé was inherently selfish, possessive. He wanted her to himself. He even accused Obi Wan and Padmé of having some sort of inappropriate relationship behind his back.
Redemption eventually came because he gave himself away for the sake of someone else. He couldn't have done this on Mustafar, because he was trying to keep what he considered his. This was jealous love, the polar opposite of the selfless love that would later allow him to redeem himself.
Lucas makes one thing absolutely clear in the quotes above - Vader's redemption was inextricably linked to his children. This alone suggests that redemption was impossible until his children existed. They hadn't been born yet on Mustafar, so redemption was still impossible.
Lucas also says that Vader was finally able to sacrifice himself because he had learned. When the events on Mustafar took place, Anakin hadn't learned anything yet. He was still too obsessed with keeping Padmé to learn from the horrific mistakes he was making at every turn.
It doesn't get any clearer than this statement, from the second quote above:
"Only through the love of his children and the compassion of his children, who believe in him, even though he’s a monster, does he redeem himself".
Nothing but his kids, and his newfound love for them, could save Vader.