If I were you, I'd go ahead and submit the transcript from your university, otherwise, you run the risk of AMCAS not accepting it, because it wasn't from your actual university if you send it via FedEx.
I'd also go ahead and submit your primary application for verification. If you submit it in early June, you can have it verified within a few days to a week. That way you won't have to worry about it taking ~6 weeks to verify during the peak application times which are during late July-Sept. And when you submit your primary, you may want to choose only one school to send it to, that way, if your MCAT score comes back as unsatisfactory, you will not be marked as a reapplicant when you apply to these schools again.