The end of August would not be too late to take the DAT, although I would try to apply by the end of July due to processing time.
I think the most overblown thing on this site is people acting like you have to apply in May. I have someone in my class that applied a week before the deadline and still got in. I think your DAT score and GPA have a lot more to do with getting in than whether you applied in May, June, July, or August.