In your notation $T$ is proper time and your $\tau$ is redundant because it equals $T\,.$ Some further notation of yours is a bit confusing. Did you mean $X=1/a$ (title) or $X=1/g$ (text)? In any case the coordinate change from <PERSON> $(T,X)$ to <PERSON> $(t,x)$ that I am familiar with is \begin{align} t&=X\sinh(\rho\,T)\,,\ x&=X\cosh(\rho\,T)\,. \end{align} The world line of the object in <PERSON> coordinates is $x^{\overline{\alpha}}=(T,X)$ where $X$ is fixed. It is known [1] that $\rho=1/X$ is the proper acceleration of the object. 563
To check this observe that the metric tensor is $$ g_{\alpha\delta}=\left(\begin{matrix}-\rho^2X^2 & 0\0&1\end{matrix}\right)\,,\quad g^{\alpha\delta}=\left(\begin{matrix}-\frac{1}{\rho^2X^2} & 0\0&1\end{matrix}\right) $$ so that $g_{TT,X}=-2\rho^2X$ and all other partial derivatives being zero. Therefore, $$ \Gamma^\delta_{\beta\gamma}=\frac{1}{2}g^{\alpha\delta}\left(g_{\alpha\beta,\gamma}+g_{\alpha\gamma,\beta}-g_{\beta\gamma,\alpha}\right) $$ becomes \begin{align} \Gamma^T_{TX}&=\frac{1}{X}\,,&\quad\delta=\beta=T,\gamma=X\,,\ \Gamma^T_{XT}&=\frac{1}{X}\,,&\quad\delta=\gamma=T,\beta=X\,,\ \Gamma^X_{TX}&=0\,,&\quad\beta=T,\delta=\gamma=X\,,\ \Gamma^X_{XT}&=0\,,&\quad\gamma=T,\delta=\beta=X\,,\ \Gamma^T_{TT}&=0\,,&\quad\beta=\delta=\gamma=T\,,\ \Gamma^X_{TT}&=\rho^2X\,,&\quad\beta=\gamma=T,\delta=X\,,\ \Gamma^X_{XX}&=0\,,&\quad\beta=\gamma=\delta=X\,,\ \Gamma^T_{XX}&=0\,,&\quad\beta=\gamma=X,\delta=T\,. \end{align} Note the factor $\rho^2$ in $\Gamma^X_{TT}$ that you missed. Therefore,
\begin{align} a^T&=\underbrace{\frac{d^2T}{dT^2}}{0}+\underbrace{\Gamma^T{TX}\frac{dT}{dT}\frac{dX}{dT}}{\frac{1}{X}\cdot 1\cdot 0=0}+\underbrace{\Gamma^T{XT}\frac{dX}{dT}\frac{dT}{dT}}{\frac{1}{X}\cdot 0\cdot 1=0}=0\,,\ a^X&=\underbrace{\frac{d^2X}{dT^2}}{0}+\underbrace{\Gamma^X_{TT}\frac{dT}{dT}\frac{dT}{dT}}_{\rho^2X\cdot 1\cdot 1}=\rho^2X\,. \end{align} Since $X$ is constant equal to $1/\rho$ we get the 2-acceleration as $$ \boldsymbol{a}=\left(\begin{matrix}0\\rho\end{matrix}\right) $$ as expected.
[1] Rindler Coordinates. 262