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No. God did not commit an injustice.
I don't see any indication that God needed or wanted the serpent to destroy this arrangement. God knew it would not last into eternity and had a plan ready for when it did collapse, but that is very different than Him instigating the fall of man. A house divided against itself cannot stand. God did not assign the serpent the task of tempting Adam and Eve.
Even now, God does not tempt a specific person to reject Him so the person can be a tool in God's plan. We all get at least one opportunity to choose Him, but there is no promise of a second opportunity. This is different than God "hardening a heart" once He has been rejected. He specifically does use people that have rejected Him to further His plans(e.g. Pharaoh)
God is just. If you discover a situation in which He appears otherwise, you certainly do not have all of the information.