I submitted on 6/3 and am taking the DAT later this week, and not worrying about supplementals until after because I've been focusing on studying.
Schools won't look at your app until they have EVERYTHING. So I figured since it's still a few weeks until my GPA is verified and another few for my scores to upload, there's no need to turn in supplementals immediately since they're not going to start evaluating it until after all that.
I don't know if this is the "best" or "correct" approach, it's just what I have chosen to do