Found a solution that works for me:

The TMP object can't be masked in 3d space as far as I can tell.

This needs to be performed simultaneously with both a sprite mask for the sprite objects and a canvas image mask for the text objects (As TMP(UI) objects). Further, the canvas image mask has to be the parent of the text objects.

To make this work, I need a very specific stacking of objects: Canvas rendered in world space, which holds the sprite objects and an image object that (at least to begin with) has nothing on it.

Then, when the card is ready to be masked, create a sprite mask and an image on the canvas image object that match, and create a mask component to the image object.

This is bulky and annoying, but it does work. I need to keep the sprite mask and the image object in-step to make the masking appear clean to the user.