A lot of people have done this. Depending on the school, it typically takes them a few weeks or a month to review your application. Generally, schools won't review your application for interview invitation until the application is 100% complete. Some schools do a preliminary look at your application without your DAT and then wait until you have your scores before inviting you to an interview.
In other words, don't expect to get any interviews until your DAT scores are uploaded. It generally takes two weeks from the date you take the exam for your scores to be uploaded to AADSAS. Add another week or two on top of that for the school to update your application status from incomplete to complete/ready for review. Then, depending on the school, it'll be a few weeks to a month before they make a decision to invite you or not for a pre-December decision (UConn for example reviews applications for interview invitation incredibly quick). Other times, the interview invitation decision process drags on to post-December.
Take the DAT ASAP. Delaying the exam does hurt you but only slightly so long as you take it before Aug. After that point, delaying the exam hurts you considerably. Why you might ask? Because if you take the exam in Sept., your application won't be reviewed until mid-October to November. By this point, the number of interview spots for pre-December decision is considerably reduced. Case Western begins inviting applicats in late-July to early-August.