... By the rules of Greek grammar, Jn 1:1c describes an anarthrous predicate, which doesn't assign Christ with the proper noun of <theos>, any more than God is assigned with the proper noun of Spirit in 4:24, and any more than Judas is assigned with the proper noun of Diabolos (devil) in John 6:70.
Again, you simply do not understand koine Greek, or John's intent in Jhn. 1:1a,b, or c; and either of the Holy Spirit which inspired the texts, since you also seem to deny His (Holy Ghost's) Personage / Being as well, and therefore cannot ever acknowledge these truths until you do (Jhn. 8:24,43; 1 Cor. 12:3).
Joh 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
Joh 1:1 εν αρχη ην ο λογος και ο λογος ην προς τον θεον και θεος ην ο λογος
As shown in the book's material, John is simply making a distinction between one Person / Being which is by eternal nature Deity (Jhn. 1:1b; ie. the Person / Being of the Father; Jhn. 1:18a,c), whom he calls "[the] God", "τον θεον" (ton theon), and the other Person / Being "with", "προς" (pros) (Gen. 1:1-31; Pro. 8:22-31; Jhn. 1:10; Col. 1:16; Eph. 3:19; Heb. 1:2, &c.) which is also by eternal nature Deity (Jhn. 1:1c; ie. the Person / Being of the Son - Jesus, Jhn. 1:17,18b), whom he calls "God", "θεος" (theos). It is not difficult to understand this, once you put away your false starting a prioric position. Daniel makes a similar distinction between the Father's Personage / Being (as "the Ancient of Days"; Dan. 7:9,13,22) and the Personage / Being of the Son (as "the son of man"; Dan. 7:13, whom is also called "angel" (Dan. 3:28; messenger [of the Father; the Father's message or word]), "son of God" (Dan. 3:25) and God (Dan. 2:47, 5:26 (connection to Dan. 8:13, the 'wonderful numberer'; ie "that certain
saint"), 11:36, &c.).
"In the beginning", "εν αρχη" (en arche) refers to Genesis 1:1's, "In the beginning", in which there is given the "elohiym" (the Godhead, or the 3 mighties; plurality, as also shown in the book's material) and the other texts already provided just above.
"was", "ην" (en), refers to the same, in that such Persons / Beings about to be described already existed before that moment, and existed at that moment. Hence, John the Baptist's sayings, and Jesus' later saying,
Joh_1:15 John bare witness of him, and cried, saying, This was he of whom I spake, He that cometh after me is preferred before me: for
he [Jesus] was before me.
Joh_1:30 This is he of whom I said, After me cometh a man which is preferred before me: for
he [Jesus] was before me.
Joh 8:58 Jesus said unto them, Verily, verily, I say unto you,
Before Abraham was, I am.
"the Word", "ο λογος" (o logos) refers to the highest messenger / message (the living Gospel [Good news] itself) of the Father, the Person / Being of the Son in Gen. 1; 1 Jhn. 1:1-3,5:7b, and contextually in Jhn. 1, as per Jhn. 1:1-3,9-11,14-15,17-18.
"and", "και" (kai), in addition to, also, moreover, so that it was not just "the Word" by Himself "In the beginning", but another is about to be mentioned alongside "the Word", directing Him about creation, for the "God said" (Gen. 1:3, &c.) is the Personage of the Father speaking to "God made" (Gen. 1:7, &c.), the Personage of the Son, there, as already shown by many texts above.
"the Word", "ο λογος" (o logos), being the Personange / Being of the Son, as previously given
"was", "ην" (en), refers to the same as previous definition.
"with", "προς" (pros) states that another exists alongside the first Personage / Being (that of the Son), being the Personage / Being of the Father, as already cited previously above, called:
"[the] God", "τον θεον" (ton theon), the Personage / Being of the Father, as per Gen. 1:1-31; Pro. 8:22-31; Jhn. 1:10; Col. 1:16; Eph. 3:19; Heb. 1:2, &c., and the immediate context of John 1:14,18a,c. The Father is the "God of gods" (Dan. 2:47, 11:36), as even Jesus calls His own Father, "my God" (Jhn. 20:17).
This means that "the Word" (an individual Personage / Being; the Son) "was with" "[the] God" (an individual Personage / Being; the Father) "In the beginning" at creation. That is two Personages / two Beings under their respective identifications by John, by the Holy Ghost (the Eternal witness, "God saw").
"and", "και" (kai), in addition to, also, moreover, so that there is more information to be relayed by John, & by the Holy Ghost / Spirit, about who this "the Word" is in nature, so that the reader is not confused.
"the Word", "ο λογος" (o logos), being the Personange / Being of the Son, as previously given
"was", "ην" (en), refers to the same as previous definition.
"God", "θεος" (theos), meaning "God" by nature, per context, and referring to the Personage / Being of the Son, identified as "the Word" of the Father, per the context of John 1.
This is John's, as well the Holy Ghost's, description of Gen. 1's "elohiym", wherein the "God said" is identified as the Personage / Being of the Father, called by John "[the] God", "τον θεον" (ton theon), and Personage / Being standing by the Father's Person / Being, carrying out the instructions of the first, is identified and called by John, and by the Holy Ghost, "θεος" (theos), "God", being the "God made" after the command of the Father. This aligns with all of the other texts in which it is continually stated that the Father made all things by / through the Son, since the Son is the ultimate Personage / Being who carries out the will of the Father, for the Father, as the Father's "Word".
John 1:1, then clearly states that:
God (Deity, the Son) was with the God (Deity, the Father), in the beginning, and God (Deity, the Son) was the Word of the God (Deity, the Father).
The Third Personage / Being, is also God, but hidden behind the scenes in Jhn. 1:1, as the Eternal Witness, who inspired John to write of those events witnessed "in the beginning", being the Holy Ghost / Spirit, the very "God saw" Whom was also present "In the beginning", taking the snapshot of events for posterity and for truth.
Deny these facts, and you cannot ever (and I mean "ever") have a proper and fullness of understanding of salvation / redemption, the purposes and events involved, or even of the proper doctrines / teachings on any subject.
Jesus, the Son is assigned the proper noun, in Hebrews (by Paul citing OT), "ο θεος" (Heb. 1:8; courtesy of Psa. 45:6) and again "ο θεος" (Heb. 1:9), and even directly having the name JHVH (JEHOVAH) applied to Him, in Heb. 1:10, 13:6(,8), citing Psa. 102:12,24-27, &c. The reason that john does not do so in John 1:1, is because he, by the Holy Ghost, is making the distinction between the two Personages / Beings which are Deity. It is the inspired commentary on Gen. 1 by the Holy Ghost. The "midrash" of Gen. 1 to use a phrase. John is simply saying that Jesus existed at the beginning, created the world on behalf of the Father's will at the Father's side, and then this eternal Son came (was sent by the Father) to dwell in the fallen flesh (Rom. 8:3) of mankind as "God manifest in the flesh", "θεος εφανερωθη εν σαρκι" (theos ephanerwthe en sarke); 1 Tim. 3:16; notice
not "τον θεον" (the Father), but rather "theos" (the Son)), and came "to his own" (His own peoples that He had taken out for Himself way back in Exodus, as "the LORD" (JHVH, JEHOVAH) "God" in the burning bush, as the "angel" (Messenger / Word) of the LORD God His Father (the other JHVH, JEHOVAH the Ancient of Days)).
You have misunderstood why John is only using the "proper noun" with the Father in John 1:1. It is simply to distinguish two Personages / Beings which are both eternal Deity and called "God". It's like distinguishing two personages / beings called "Boss", both of which are of the nature humanity. You also do not seem to understand how koine Greek works, and translation into English. There are plenty of places in the GNT TR where the definite article is used (ie Jhn. 1:35, "ο ιωαννης" (o iwannes), literally "the John"), and not translated into English, as English does not really work that way too well, and would make for a more difficult reading to the English reader, and places in the GNT TR in which a proper noun is not given, and simply 'anarthous' and yet is translated into the English with the definite article "the" (Jhn. 1:1,18, 3:16, 8:12, 10:7, &c.), since the English would require it in such places to read smoothly, rather than harshly.