The error being made in using Genesis 3:1 to say God did not make the serpent is one of incorrectly parsing the genitive phrase. In order to make the case (and thereby introduce inconsistency with other scripture, you would need to prove that:
- "of the field" is less important than "which the Lord God had made" and only random in order
- that "which the Lord God had made" modifies beasts but not Serpent.
Absent these proofs, the claim implicit in the question (that the Serpent is not a beast which God made) is simply not supportable.
The verse, of course, says:
Now the serpent was more subtle than any beast of the field which the Lord God had made
To make your case, you need to emphasize the which the Lord God had made:
Now the serpent was more subtle than any beast
of the field(thing) which the Lord God had made
Or, put another way
Serpent > Any (thing) God made.
This would be a valid interpretation if this were to be treated as a singular genitive phrase.
By contrast, the more mainstream reading would be
Now the serpent was more subtle than any beast of the field which the Lord God had made.
In this reading, the emphasis is on the phrase 'any beast of the field'. The idiomatic expression 'which the Lord God had made' is a subordinate detail to beast of the field - it is simply a modifier. Put another way,
Serpent > Any beast of the field. Incidentally, God made the beasts, and we are scoping this as all beasts that God made, which is all of them.
In order for Case 1 to be valid, other references (especially John 1:3 - All things were made by Him, without Him was nothing made that was made) would be in conflict. This alone would be enough for a coherent reading of all Scripture to make the point - but, I can, for sake of argument, not take this tack.
Instead, one would need to look at emphasis and word order, which in Hebrew and English is significant. The mere fact that beasts is modified by two prepositional phrases should already hint at where the emphasis would lie.
Furthermore, it is equally valid, that as a genitive phrase, which the Lord God made could be modifying both serpent and beast. It would be syntactically ridiculous to say:
Now the serpent which the Lord God has made was more subtle than any beast of the field which the Lord God had made
but it would be expressing the equally logical thought. In order to prove Case 1, you would need a compelling reason to discount this reading, which of our course cannot really made.