What you describe there is commonly known as "cancer".
I think cancer will be cured much sooner than there will be a working teleportation technology(for example just by storing the DNA and replacing the mutated DNA using nanotechnology).
Also the flaw you are describing(10⁻⁹(where did you get this number?)) is big. The error could lead to a displacement of your atoms of about 2(2 m)10⁻⁹ = 4 nm which is huge. The cell membrane is about the same size and would therefor get destroyed and you would die. Until the flaw would be much less(10⁻¹⁰ - 10⁻¹¹) this technology would never get used.
Yes changes may also affect the brain, but since the brain has an incredible power of self repair I don't think this would be an issue. Especially not with flaws only in the region of several 100 picometers.