Does the Dawn Prayer become invalidated if the worshipper recites the Istikhara supplication after completing it?

This supplication has no relation to your prayer and does not invalidate it. Rather, your prayer is valid, and you are not required to repeat it. This is from the whispers of Satan, so do not pay attention to it. We advise you to seek knowledge and learn what is obligatory and what is required for prayer, and what invalidates it, so that you may have clear insight in your worship.