I've heard horror stories of guys escorted out of the building for pretty much the same thing. "Your application dated xx/xx/xx said you had a degree, but your diploma and graduation record says you got your degree 2 months later. Tell us about that." If you say you have something in your application but you didn't actually have it until 2 months later, then they very well may conclude that it was a lie on the application and you'll never work there, ever.
Why risk getting blacklisted to get one question on your app updated 2 months early? Don't ever do that on an application, ever, even if somebody at a job fair says yea go ahead. The question is can you survive being asked why you put down an answer at a particular time, when the answer was not verifiable as true at that time or when written documentation will show that it was not in fact true, and you knew it.
This goes a bit deeper than accidentally clicking submit when your class 1 medical is a month overdue.
All that said, why not contact the registrar and see if they can graduate you early? I had to do that to get an advanced degree included in some paperwork, to the point where I skipped the term project and took a B in my final class and threw away a 4.0 GPA because the paperwork submission date had been moved up by over a year with very little notice so I couldn't wait until the end of the term. I had to be able to prove I had an advanced degree right away, so I told the teacher I was done submitting work and could I have my diploma immediately, pretty please. Less than a month later I had a final transcript and diploma. It turned out it didn't matter because of reasons, but the point is that some colleges will understand your situation and may graduate you immediately if you ask the right person.