Based on what others are saying on SDN the current AACOMAS verification time is 3-4 days.
To add onto what others said, some schools will even accept your secondary application before you receive an invite. DCOM told students at my university, for example, that as soon as we hit submit on the AACOMAS application that we should go to the website, download the secondary, and send it in. There are others who say the same. Thus, at schools like DCOM, you could be entirely complete with the exception of the letter from the DO. Further, since DCOM doesn't require a DO letter, if you've also got an MD letter you'd be entirely complete.
Get everything as complete as you can get it as quickly as you can get it. Don't rush your essays but there's no reason not to shotgun everything else (LORs, etc) as fast as you can. The less pieces of the puzzle missing the better!