Upon arrival in FRA early morning go to the AiRail counter (follow signs for Terminal T which is the long distance train station) and ask if you can take an earlier train with your boarding pass - sometimes they will rebook you without charge. The train also stops in Mannheim and you can leave the train there. Remember that you have to collect your luggage labeled for ZWS (Stuttgart train station) in FRA at the AiRail area and take it with you onto the train by yourself.

Nothing will happen, if you don't take the booked train to Stuttgart but instead use another form of transportation or buy a separate train ticket to Mannheim for an earlier train. Train connections will be at least once per hour nonstop between FRA Flughafen Fernbahnhof and Mannheim. Bus might be a few Euros cheaper but (especially with luggage) a bit less comfortable.

For your journey back to FRA you'll have to check in for your return flights before checkin deadline of your train to Frankfurt. You can do that online up to 23h before or in person at the AiRail area in Stuttgart Hauptbahnhof. Again nobody will cancel your ticket if you're not sitting on your seat - you could have been to the restroom or the restaurant area... ;-)
But you HAVE TO BE CHECKED IN for the whole trip before the deadline. Checking in in FRA only without prior online checkin for the trip will result in problems or cancellation.
If you do an online checkin you can also board the train just in Mannheim heading to FRA with your online BP.
Your luggage for the ongoing flight from FRA can then be left at the AiRail counter in FRA right at the train station.