What is the Biblical Basis for teaching that Lucifer wanted to become the “firstborn son” of God through Eve?
Lucifer desire to be like God in the Book of Isaiah but it does not explicitly said that desiring to be like God is equal to be desiring to be like the Incarnated Logos or the "firstborn son" of God.
“I will ascend to the heavens; I will raise my throne above the stars of God. I will sit on the mount of assembly, in the far reaches of the north. 14I will ascend above the tops of the clouds; I will make myself like the Most High.” (Isaiah14:14)
However, Lucifer knew God's Divine Plan in the Heavenly Realm about the Logos Incarnation that the Logos will be the "firstborn son" of God thru a Woman.
Maria de Agreda relates that when Lucifer learned that the Logos would become man through a human mother; Lucifer, the highest of all creatures, demanded the honor of becoming the Theotokos. He wanted the hypostatic union to occur through him. This is another reason why there is perfect enmity between Satan and Mary (see Gen 3:15). It is also why Mary now has the highest place in Heaven. - How did Lucifer Become Satan?
The "firstborn son" or the Logos will be incarnated thru the Woman. The Logos will be born in a form of a child, the fruit of the Woman's womb.
How did the "firstborn son" came into the world.
- The Woman or Mary conceived first the coming of the Logos in her heart. Mary had fully accepted the Will of God in the coming of the Logos.
“Mary is blessed beyond all women because she received the Son of God in her body; but she is even more blessed because she first received the Word of God in her heart.” (St.Augustine of Hippo)
- When Mary had perfectly conceived the Logos in her immaculate heart, the Holy Spirit "fecundates" her pure womb to form Jesus.
485 The mission of the Holy Spirit is always conjoined and ordered to that of the Son.122 The Holy Spirit, "the Lord, the giver of Life", is sent to sanctify the womb of the Virgin Mary and divinely fecundate it, causing her to conceive the eternal Son of the Father in a humanity drawn from her own.
The Holy Spirit "fecundated" or "impregnated" the womb of Mary.
- The Holy Spirit fecundated Mary's womb without a human seed.
From the first formulations of her faith, the Church has confessed that Jesus was conceived solely by the power of the Holy Spirit in the womb of the Virgin Mary, affirming also the corporeal aspect of this event: Jesus was conceived "by the Holy Spirit without human seed". (Council of the Lateran (649): DS 503; cf. DS 10-64.)
The "firstborn son" of Mary the chosen Woman and the New Eve came into this world without a human seed.
Can Lucifer replicate the work of the Holy Spirit in the pure womb of Eve to produce a "firstborn son" thru his works or power? Anyway, Lucifer does not need a "human seed",can Lucifer fecundate the womb of Eve?
But before Lucifer can imitate the Holy Spirit and tries to fecundate Eve, one thing must happen first. Like Mary, Eve must conceived first the "firstborn son" in her heart, Eve must willed and must fully desire that she wanted to have a "firstborn son".
Did Eve fully desire to have a "firstborn son"?
This is where the theory of "serpent seed" came into play.
Serpent seed, dual seed or two-seedline is a controversial religious belief which explains the biblical account of the fall of man by saying that the serpent mated with Eve in the Garden of Eden, and the offspring of their union was Cain. It appears in early Gnostic writings such as the Gospel of Philip (c. 350). Irenaeus (c. 180), an early church father, explicitly rejected the doctrine as heresy, a view which was echoed by subsequent mainstream Christian theologians. According to The Celtic Church In Britain, by Leslie Hardinge, the early Celtic church taught the belief in re-telling the seduction of Eve by the serpent.
The idea that Eve mated with the serpent, or Satan, and produced Cain, finds its earliest expression in Gnostic writings (e.g., the Gospel of Philip). It was rejected by mainstream Christian theologians such as Irenaeus. The idea appears in a 9th-century book titled Pirke De-Rabbi Eliezer. In his book Cain: Son of the Serpent, David Max Eichhorn traces the idea back to early Jewish Midrashic texts and identifies many rabbis who taught that Cain was the son of the union between the serpent and Eve. - Serpent seed (Wikipedia)
Obstacles to Lucifer desire to become the "firstborn son";
Can Lucifer fecundate or impregnate the womb of Eve like the Holy Spirit did to Mary? Yes or No?
The answer is No, Lucifer is only a created being and not a Creator much more Lucifer cannot give life to a creature like the Holy Spirit did to Mary.
The obstacle to Lucifer fecundating Eve is, Eve did not fully desire to have a "firstborn son", the scriptures did not explicitly mentioned Eve desired to conceived a "firstborn son" unlike Mary the chosen Woman had conceived it in her heart.
The mystery question is, did Lucifer tried to "fecundate" the pure womb of Eve?