In addition to the points already provided, may I offer a more obvious point based on simple logic?
So, the question is, should the latter θεός in John 1:1 be translated into English as "God" or "a god"?
In John 1:3, it is written that «πάντα δι᾽ αὐτοῦ ἐγένετο καὶ χωρὶς αὐτοῦ ἐγένετο οὐδὲ ἕν ὃ γέγονεν», that is, "All things were made by him, and not one thing that was made was made without him."
Now, folks, all things were made by the Word (ὁ λόγος, which is the antecedent of the pronoun αὐτοῦ in John 1:3).
If the Word is "a god" but not "God," how then did the Word "make all things"?
"A god" is created (a creature), for only God is uncreate and eternal. Only God is the creator of all things.
So, how did a god create all things when he himself had to be created?
It's illogical. No, it's not a "mystery" that we can simply brush off. It's illogical. It's a contradiction. The Bible is not a book of contradiction. It's a book of truth. The Word must be God because it created all things.
Simple as that.
If the Word is the creator of all things,
And the creator of all things is God,
Then the Word is God.
Again, simple logic.
If θεὸς was definite (arthrous), then it would mean that the Word was the Father.
If θεὸς was indefinite (anarthrous), it would mean that the Word was "a god" --- essentially promoting polytheism. Add that to the fact that we are said to honor the Son as we honor the Father (John 5:23), and you'd be committing idolatry by worshipping "a god" rather than the only true God, YHVH.
Rather, θεὸς is neither definite, not indefinite, but qualitative.
Let me provide examples using the word ἄνθρωπος.
Suppose I wanted to translate the following English phrases into Greek:
"He is the man."
αὐτός ἐστίν ὁ ἄνθρωπος.
"He is a man."
αὐτός ἐστίν ἄνθρωπος.
Both of these phrases tell me who the man is. The former tells me that he is a certain man, as noted by the definite article ὁ preceding ἄνθρωπος. The latter tells me that he is a man, but it does not designate a particular man.
But, suppose I wanted to tell you what he (αὐτός) is? Not who, but what.
In his Categories (1.5), Aristotle writes, "For instance, the individual man is included in the species 'man', and the genus to which the species belongs is 'animal'; these, therefore that is to say, the species 'man' and the genus 'animal, are termed secondary substances."
οὐσία describes what something is (without getting into a lengthy philosophical discourse), not who it is. And, Aristotle states that ἄνθρωπος qualifies as a "secondary οὐσία."
Now, some people may think, "Aristotle, really?" Well, some people may think that John was an ignorant fishermen, but at the least, he had an exceptional grasp of the Greek language. In fact, I believe he was remarkably learned and intelligent --- especially in philosophy. This is, after all, the same man that begins his Gospel with the mention of one of the richest philosophical terms in history, λόγος. John also wrote to Greeks, and isn't it obvious that Greece was the epicenter of philosophical thought?
In summary, when John writes, «καὶ θεὸς ἦν ὁ λόγος», he is telling us what the Word is --- its nature --- not who the Word is (i.e., the Father). Like ἄνθρωπος, θεός can also refer to an οὐσία.