An Angle-Bisector Foot

Let $O$ be the origin and let $F_1, F_2$ be the left and right foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with eccentricity $\dfrac{\sqrt{6}}{2}$. Let $P(x_1, y_1)$ be a point on the right branch other than the vertex. The perpendicular from $F_2$ to the bisector of $\angle F_1PF_2$ has foot $M$, and $|MO| = \sqrt{2}$. If a point $T$ on $C$ satisfies $\overrightarrow{F_1T}\cdot\overrightarrow{F_2T} = 5$, find the sum of the distances from $T$ to the two asymptotes of $C$.