Found a workaround: Switch the game to windowed mode and resize the window until it is small enough that the framerate is smooth. After that, the cutscene works perfectly. Only problem is that now the dialogue window is too small to be able to read the whole text, so this is only a temporary solution until a patch for us poor people with low-end graphics is released.
Also, I couldn't find where the game puts the log file in my Linux GOG install. It doesn't seem to be anywhere...