I've developed a solution that has been working in the office for a few months now.
It's maybe not perfect, but elegant enough, does the job reliably and is 100% stable.
1 - In title block Family create the Date/Time Stamp Label.
2 - Drag the controls of that Label narrower, so that Date and Time split into two lines. Date is not on top, Time below that.
3 - Create a new Tag Label Tag, the same Type Properties as the one in in first step, but give it Color, that is just off white. Say 254,255,255.
4 - Can you see the solution already? Yes - Create an Instance of that off white label, where the second line of the normal label is in position and Horizontal Align value that will cover up the Time. The time from the white label covers up the time from the black label, while the white date is not visible - white text on white paper.
As you can see, while neither the Filled Region or the Masking Region actually masked the Tag, another tag, that is created later, will.