Why discretize a continuous transition matrix in Kalman Filter?
Have you taken a look at the documentation - section 2.2.4 discusses a Linear Kalman filter model that is very similar to the one you described. From that example you see that:


The resulting $\bf{A}$ matrix does not depend on the process noise
The $\bf{A}$ matrix only depends on the size of the time step.
The $\bf{Q}$ matrix only depends on the size of the time step, and the value of the continuous noise covariance, ie. diag(q) in the example


You can use discrete noise kinematic models, they are probably a bit easier to understand. The continuous noise model assume that the noise is applied throughout the time step interval i.e. from time $t_0$ to $t_1$. The discrete model assumes the noise is applied instantaneously at times $t_0$ and $t_1$ (and so on).

See chapter 6 -"Estimation for Kinematic Models" in the book - "Estimation with Applications to Tracking and Navigation" (Bar-Shalom, Rong Li, Kirubarajan) for a derivation of the continuous and discrete time versions of the state transition models.