Keep in mind that if you submit in early June, it takes a few days to have your transcripts verified. By the end of June, it can take a few weeks. By August it can take a month or more. So it's better to take the class that is done in May, if you need to submit the grade to improve your application GPA. Otherwise, just list it as a future course.
I thought it usually took about a month for an MCAT score release. Have they changed it to six weeks? Anyway, a July release date wouldn't be a huge issue if you've already submitted to AMCAS earler and completed the verification process. Then, when the MCAT score is released, your entire application gets to all schools within 24 hours. If you are not confident of your MCAT score, you can initially apply to only one school and add others later if your score is competitive. Newly added schools also get the entire file within a (business) day.
The August release date would not be one's first choice considering your chances of getting an interview decrease as more time passes. But if you had no choice, assuming preverification as above, and prewitten Secondary essays (from prompts here on SDN), you could get all your Secondaries completed and returned within a few dates of receiving them and still get a reasonable degree of consideration.