We have grown a complex system with multiple NDI outputs for projection stage, and several overlays. We have repurposed some text styles to show or hide on each display. For example, we have used text styles A, B, and C as:
A - 'All' display on all, highlighted text on screen for congregational response
B - 'Both' display on all, normal text for reading for all
C - 'Call' display on stage only for leaders

Is there any way to tag text as a certain style when importing using Text Slides Assistant? Alternatively, is the slide show file format published anywhere so we can write using our own tools?

John,

There are currently no ways to tag text to display in a certain style when importing.

There are currently no code/API tools available to allow users to create tools for SSP.

It sounds like you are using the some of the type specific styles.  You could further leverage this by using themes and templates when you create your content.  The Assistants don't have any options to create or select existing themes or templates but you can edit that after creation.  Also you can change the slide object styles after creation.  You can contact the support team to further explain the intended outcome to see if they can provide a more efficient way of getting the result you are looking for.