How can I trigger an intercept based on a value in the data layer for a GTM push?

We don't do that specifically, so I can't speak from experience, but under intercept display logic, you can add a condition based on event tracking (under website). The intercept won't display until it's set to under the display options (on page load, on exit intent, or manual rules) but with the display logic, it would of course require the event to take on whatever value you define first.