I would say, that the question is not put entirely correctly, for it contains an implicit information, with which I (and anybody) may not agree from the outset; namely, the question implies that "Son" is, or can be, Logos' name only after the Incarnation (when the Logos was called also Jesus), but not prior to the Incarnation, so as to affirm that even in eternity, even before the world was created, the Logos of the Father was simultaneously the Son of the Father.
However, this is not even a moot point, for the eternal Logos is also the eternal Son, even before the Son's Incarnation. How otherwise indeed?! Is not God eternally the Father? Yes, He is! Even before creation of the universe? Surely, even before creation of the universe, for 'eternally Father' means that creation of the universe, which exists not in eternity, but in time, does not introduce fatherhood in Him. Now, if He is eternally the Father, then given that the universe does not yet exist, He must not be called Father with reference to anything created, must He? No, He mustn't. But to be father means to have son (or daughter, but let us forget for a while this political correctness nausea, for it adds nothing to argument), yes? Yes! And thus the eternal Father must have co-eternal Son, mustn't He? Yes, He must indeed and necessarily so. But is not it acme of absurdity to affirm that Father has co-eternal Logos and, besides, co-eternal Son also, and + Holy Ghost who proceeds From Him (John 15:26), so that we get not Trinity but Quaternity of Father the Son the Logos and the Holy Ghost. But let us send to the world of phantasy this idea and return to a sound theology which identifies Logos with the Son. For also Jesus calls the One with Whom He enjoyed Glory before even the creation of the universe "Father" (John 17:5), and since it is contrary to any sound reason to think that God became His Father only after He adopted created human nature, then we must necessarily assume that He was the Father's Son already in eternity before the creation of universe, to say nothing before the Incarnation. In fact, when He says "I came from the Father and entered the world" (John 16:28) does not this show without any taint of ambiguity that the Father was His Father even before the Incarnation, for the Incarnation means 'entering the world', while even before that He came out of Father, to the effect that the Incarnation has nothing to do with His and Father's eternal Fatherhood-Sonship.
Therefore, yes, it is a perfectly good logic that you bring: if eternal God's co-eternal Logos is God and if this Logos is also Son, then necessarily the Son is also God.