"the serpent ... is the Satan himself"
I think any other view is untenable.
Consider the following Scriptures:
Genesis 3:15 (DRB) [brackets mine]
And the Lord God said to the serpent: Because thou hast done this thing, thou art cursed among all cattle, and the beasts of the earth: upon thy breast shalt thou go, and earth shalt thou eat all the days of thy life. 15 I will put enmities between thee and the woman, and thy seed and her seed: [the same] shall crush thy head, and thou shalt [strike the] heel thereof.
Now in the Old Testament, the 'types' or shadows of the coming fulfillment has the woman crushing the head of the enemy of Israel (because it was through a woman he sought to doom us), typifying Satan, the perennial enemy of mankind. The consistent theme always being the enemy of God's people being attacked with a fatal head injury, by a heroic woman:
Blessed among women be Jahel the wife of Haber the Cinite, and blessed be she in her tent. 25 He asked her water and she gave him milk, and offered him butter in a dish fit for princes. 26 She put her left hand to the nail, and her right hand to the workman's hammer, and she struck Sisara, seeking in his head a place for the wound, and strongly piercing through his temples. 27 At her feet he fell: he fainted, and he died: he rolled before her feet, and he lay lifeless and wretched.
And by me his handmaid he hath fulfilled his mercy, which he promised to the house of Israel: and he hath killed the enemy of his people by my hand this night. 19 Then she brought forth the head of Holofernes out of the wallet, and shewed it them, saying: Behold the head of Holofernes the general of the army of the Assyrians, and behold his canopy, wherein he lay in his drunkenness, where the Lord our God slew him by the hand of a woman. 20 But as the same Lord liveth, his angel hath been my keeper both going hence, and abiding there, and returning from thence hither: and the Lord hath not suffered me his handmaid to be defiled, but hath brought me back to you without pollution of sin, rejoicing for his victory, for my escape, and for your deliverance. 21 Give all of you glory to him, because he is good, because his mercy endureth for ever. 22 And they all adored the Lord, and said to her: The Lord hath blessed thee by his power, because by thee he hath brought our enemies to nought. 23 And Ozias the prince of the people of Israel, said to her: Blessed art thou, O daughter, by the Lord the most high God, above all women upon the earth. 24 Blessed be the Lord who made heaven and earth, who hath directed thee to the cutting off the head of the prince of our enemies. 25 Because he hath so magnified thy name this day, that thy praise shall not depart out of the mouth of men who shall be mindful of the power of the Lord for ever, for that thou hast not spared thy life, by reason of the distress and tribulation of thy people, but hast prevented our ruin in the presence of our God. 26 And all the people said: So be it, so be it.
And Abimelech coming near the tower, fought stoutly: and approaching to the gate, endeavoured to set fire to it: 53 And behold a certain woman casting a piece of a millstone from above, dashed it against the head of Abimelech, and broke his skull. 54 And he called hastily to his armourbearer, and said to him: Draw thy sword, and kill me: lest it should be said that I was slain by a woman. He did as he was commanded, and slew him.
We would be mistaken not to see the theme here: woman crushes head of enemy, which brings temporal salvation to God's people.
Go to the New Testament, and of course the prime candidate is the mother the true Seed, Jesus Christ:
Luke 1:26-35, 29-55
26 And in the sixth month, the angel Gabriel was sent from God into a city of Galilee, called Nazareth, 27 To a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary. 28 And the angel being come in, said unto her: Hail, full of grace, the Lord is with thee: blessed art thou among women. 29 Who having heard, was troubled at his saying, and thought with herself what manner of salutation this should be. 30 And the angel said to her: Fear not, Mary, for thou hast found grace with God. 31 Behold thou shalt conceive in thy womb, and shalt bring forth a son; and thou shalt call his name Jesus. 32 He shall be great, and shall be called the Son of the most High; and the Lord God shall give unto him the throne of David his father; and he shall reign in the house of Jacob for ever. 33 And of his kingdom there shall be no end. 34 And Mary said to the angel: How shall this be done, because I know not man? 35 And the angel answering, said to her: The Holy Ghost shall come upon thee, and the power of the most High shall overshadow thee. And therefore also the Holy [One to be] born of thee shall be called the Son of God.
39 And Mary rising up in those days, went into the hill country with haste into a city of Juda. 40 And she entered into the house of Zachary, and saluted Elizabeth. 41 And it came to pass, that when Elizabeth heard the salutation of Mary, the infant leaped in her womb. And Elizabeth was filled with the Holy Ghost: 42 And she cried out with a loud voice, and said: Blessed art thou among women, and blessed is the fruit of thy womb. 43 And whence is this to me, that the mother of my Lord should come to me? 44 For behold as soon as the voice of thy salutation sounded in my ears, the infant in my womb leaped for joy. 45 And blessed art thou that hast believed, because those things shall be accomplished that were spoken to thee by the Lord.
46 And Mary said:
My soul doth magnify the Lord.
47 And my spirit hath rejoiced in God my Saviour.
48 Because he hath regarded the humility of his handmaid;
for behold from henceforth all generations shall call me blessed.
49 Because he that is mighty, hath done great things to me;
and holy is his name.
50 And his mercy is from generation unto generations,
to them that fear him.
51 He hath shewed might in his arm:
he hath scattered the proud in the conceit of their heart.
52 He hath put down the mighty from their seat,
and hath exalted the humble.
53 He hath filled the hungry with good things;
and the rich he hath sent empty away.
54 He hath received Israel his servant,
being mindful of his mercy:
55 As he spoke to our fathers,
to Abraham and to his seed for ever.
This when we come to the fulfillment of the types or shadows, when through a woman God brings eternal salvation, and defeat of Satan: in the form of a new Adam and a New Eve—a new woman who is markedly not like the Eve of whom she is the replacement, and counterpart in the New Creation.
But in Scripture we also see it made quite explicit in a few places that the decieving serpent was Satan himself:
For God created man incorruptible, and to the image of his own likeness he made him. 24 But by the envy of the devil, death came into the world: 25 And they follow him that are of his side.
Specifically, with regard to Satan being crushed under the feet of the woman and her seed, harkening back to this prophecy:
And the God of peace crush Satan under your feet speedily. The grace of our Lord Jesus Christ be with you.
Notice that the Seed (which is simply a Hebraism for "offspring") of the woman is Christ, but also His followers, as seen from the following Scriptures:
And a great sign appeared in heaven: A woman clothed with the sun, and the moon under her feet, and on her head a crown of twelve stars: 2 And being with child, she cried travailing in birth, and was in pain to be delivered. 3 And there was seen another sign in heaven: and behold a great red dragon, having seven heads, and ten horns: and on his head seven diadems: 4 And his tail drew the third part of the stars of heaven, and cast them to the earth: and the dragon stood before the woman who was ready to be delivered; that, when she should be delivered, he might devour her son. 5 And she brought forth a man child, who was to rule all nations with an iron rod: and her son was taken up to God, and to his throne [the Ascension] . . . that old serpent, who is called the devil and Satan, who seduceth the whole world ... And the dragon was angry against the woman: [cf. Gn 3:15a] and went to make war with the rest of her seed, who keep the commandments of God, and have the testimony of Jesus Christ.
Now there stood by the cross of Jesus, his mother, and his mother's sister, Mary of Cleophas, and Mary Magdalen. 26 When Jesus therefore had seen his mother and the disciple standing whom he loved, he saith to his mother: Woman, behold thy son. 27 After that, he saith to the disciple: Behold thy mother. And from that hour, the disciple took her to his own.
I believe given Revelation 12, Wisdom 2, etc. that his followers, that is, Satan's, including "his angels," who rebelled with him and were cast out of heaven are his "seed." Just as the spiritual children of Mary, by way of brotherhood with Christ, (Heb 12:11) are her "seed."