Call the school and see if there's an appeal process for entering kids whose birthdays are shortly after the cutoff.
My state (Maryland) has a 9/1 cutoff. But both of my kids have students in their classes whose birthdays are just after the cutoff. So there's some kind of appeal process for it here.
The other option would be to look for a private school that has half or full day K and have your child do K at a private school, then do another year of K in public school when he meets the age requirement.
We've moved a lot (not military, though) and found ourselves in a similar situation. DD has an October birthday. She started preschool at almost-3 in a state with a December cutoff, and then in the middle of her second year of preschool we moved to a state with a firm August 31 cutoff for kindergarten and first grade. I didn't research the public schools very thoroughly because it was clear there wasn't going to be any getting around it, especially since she was a month and a half past the cutoff. But I felt strongly she was on track to be ready for kindergarten in the fall. I selected a pre-K program at a Catholic school based partly on their willingness to promote her to kindergarten that fall pending assessment by the pre-K and kindergarten teachers, who agreed she was ready after she had been there a few months, so she started kindergarten last fall. In January we moved again (!) to a district with a September 30 cutoff, and the principal at the nearest public school agreed to let her continue in kindergarten on the understanding that they could have her repeat K if necessary. But she has done just fine and will be continuing on to first grade this fall.
Now when I say my daughter was ready, I mean I was dead certain she was ready in terms of academic preparation and classroom behavior and 90% sure she would be fine socially, although she started out preschool rather shy (she's not anymore, though). She has been resilient through our moves, makes friends easily, and adjusts well to new classroom environments. A lot of parents would still have gone with the local cutoff on the assumption that being one of the oldest in her class instead of one of the youngest would be preferable. And while I definitely believe, especially now that the year is nearly over, that we made the right decision, I do think girls tend to mature earlier and that the disadvantages of being younger than her peers (size, for example), are not as pronounced. I also admit that since DH and I were both among the younger students in our classes but still did very well academically, I had similar expectations for DD, and risked projecting my experience onto her unique personality. (I think she is bright but I wouldn't venture to say yet that she's gifted.) So, while in general I advocate determining kindergarten readiness by individual assessment rather than birthdates, you should be certain that this is the right step for your son and expect that you will encounter people who are either surprised you want him to start kindergarten early or discourage you from doing so. I can't tell you how many times I've heard people say that nobody regrets holding their kids back but many regret starting their kids in kindergarten too early. All I can say is that so far I don't regret it. But be certain you won't either.