I wouldn't push it past mid-July if you don't want your application to be considered late. As far as hard date cutoffs go, they vary from school to school.
so it takes 2-3 weeks from the time you take the DAT until AADSAS gets it and your application is submitted.
It takes around 1 month for your transcripts to be converted by AADSAS into their standardized GPA. They don't do this until you submit your application (i think). So i think submitting the app by the end of july would be your best bet (that would require you to take DAT mid july the latest)