After a little bit of digging in the game files, there is a PlaySound function.
What I have done so far use a TTS program to generate sound files of a couple of lines of the dialog and place the .wav files (for simplicity I have labelled them the same as the text string name that it is replacing eg Mission1_t0_0a.wav) into a new folder in the Assets\Audio, called "Dialog".
I have then added an additional lines to the UI_Game.audiogroup file (comma separated text), eg:
(I have not experimented yet if the game would load additional audiogroup files from the folder, or if tampering with an existing one is needed.)
Then using <PlaySound Sound="Mission1_t0_0a"/> at the appropriate trigger should play the sound file. With a bit of tweaking to the volume levels, and hack the hell out of the mission triggers, I believe that it is feasible.