Two weeks can absolutely make or break your score on the DAT (or it could make no difference at all).
Personally, I scored close to the same scores for a month of practice tests (and/or sections), until the last couple of weeks when it all began to click and come together.
exes, the bottom line is this:
Schools can overlook a weaker DAT with that kind of GPA, but only to a certain point. Score as well on your DAT as you can. Believe me when I tell you two extra weeks for a completely finished application is well worth it if you can go from a (for example) 17 -> 19. Id even advocate maybe a month to go from a 17 -> 20, especially if you need to devote some extra time for the reading section.
Additionally, submitting your application that early I believe usually takes AADSAS a little longer since they have such a high volume of submissions at one time. It will probably take them a month to send them to schools. It takes them less time (i think approx 2 weeks) to receive, validate, and send DAT scores to schools. So, even if you took your DAT a month later (in beginning of July), assuming you submitted your application on June 4th (the day the cycle opens), you really would only be delayed a couple of weeks and still WAAAAY ahead of most applicants.
Again, my advice would be to wait until you are comfortable taking it! Be realistic with yourself and where you are at. The worst mistake you could make would be to rush to take it, do bad, and have to wait 3 months to take it again for a decent score- then you would REALLY be late!