This is the same type of question people ask about other deviancies, like homosexuality. The issue of transgendered people, along with homosexuality are the result of man's fallen nature in the sight of God. So, no, God doesn't make the mistake, sin causes the mistake.
Glad to hear you say that!
Now what of the question... should we be living as we were born?
__________________ Mrs. LPW
Psalm 19:14
Let the words of my mouth, and the meditation of my heart, be acceptable in thy sight, O LORD, my strength, and my redeemer.