**(1)** You are using an incorrect expression for acceleration which is valid for classical cases only. That is, $\ a = \frac{\ v^2}{\ x}$ . The correct one is $\ a = \frac{\ v^2 \gamma}{\ x}$

These pages from Hartle's book Gravity will help. See example 5.6, equation 5.52 and whats written after it.

![enter image description here][1]

![enter image description here][2]

  [1]: https://i.sstatic.net/EuVJj.jpg
  [2]: https://i.sstatic.net/VuKbx.jpg
**(2)**  $\ dt$ has to be corrected for. Use $\gamma\ {dt_{rest frame}}=d\tau_{rest frame}$ See the line following equation 11.119 in your textbook.