Well you would get your results in October which is not late for DO schools, but it also depends on your stats because if you by chance have below average stats, it is not always ideal to be applying later in the cycle. However, you could have you application verified and ready to go so when your MCAT scores come in it would be ready to send out.
I’m not one to insist that an application needs to be submitted in June and July but a September 28 test date is too late. Your scores would come back October 13. Assuming you submit AMCAS for verification early, you would need to finalize your list of schools post mcat score, wait for secondaries which could be anywhere from one day to several weeks, and then submit secondary applications. Even assuming you have prewritten all secondary essays, the timing simply isn’t in your favor. Now, schools may shift interview dates later in the cycle due to AMCAS and mcat delays, but your app would be on the very tail end of the cycle.
I would consider either moving the test date up to mid August or plan on applying next cycle. I’m not one to freak out over applications being completed in June and July. Generally speaking, before Labor Day is early, mid to late September is normal, October is late.