Obviously, taking it any time later than the end of May/beginning of June will delay your app, and as a result will hurt you. The issue is whether or not you will be prepared for the MCAT by the end of May/beginning of June, because as others have stated, a low MCAT score will hurt you more.
If I were you, I wouldn't schedule the MCAT for later than June. Personally, I don't think you need to study for 3 months, but everyone is different so only you can figure that out. 1 month worked just fine for me, and I think I would have reached my peak at 1.5 months. If you are able to multi-task and study during school, you can still fit in lots of MCAT prep time.
If you have the extra $75 (cost of changing the test date), I'd schedule it for around the earliest you think you'll be ready. If by 2 weeks before, you don't feel confident and are not getting at or above your target on practice tests, I'd postpone it to a slightly later date.
Regardless of when you take it, make sure your app is in and verified, you have all your LORs, and have secondaries pre-written by the time your MCAT score is released. This is all provided you have a GPA >3.0.