Never lose hope until you've truly talked to everyone everywhere because you just never know. Don't necessarily get your hopes up too high, but it's always worth a shot.
My DD8 just missed the cutoff here of Sept 1 (8 days). I can't tell you how many times we were told by various people throughout the district that there were no exceptions to the deadline. Over time we just kept asking various people, from the office staff at the elementary school to the screeners at the k-screening (after she blew the eval out of the water) to eventually the superintendant, who told us it was district policy but truly up to the school principal. We heard a lot of "Sure she can read, but that doesn't make her ready for kindergarten." We knew it was much more than being able to read so we persevered. We had a back-up plan, though, to attend a private K and enter public 1st if need be, which is allowed here.
Once we got the attention of the school principal through a letter and phone call, we had to prove she wasn't just "another kid whose parents thought she was smart." We had my DD's Montessori preschool teacher provide her "report card" and a letter stating she was absolutely ready for K. My DD had to go into the elementary school and be observed in a k class as a visitor for an hour or so. We had to fill out a parent assessment form. We had to meet with the principal, a k teacher, and the school psychologist to discuss their concerns (they wanted to make sure that she'd had school experience previously, and that we understood all the ramification of being the youngest yada yada yada). And, it turns out -- although we didn't know it at the time -- during her time meeting with the school psych, she was administered the WPPSI and an achievement test. They were very thorough. I don't blame them for wanting proof. They do have to worry about "if we do it for one we have to do it for everyone who asks." Turns out she was the first child ever admitted early in our district. They said it couldn't be done, but when we showed them she was ready, they eventually saw the need. Two years later she was grade-skipped again.
This year my DS5.5 was scheduled to go to K but we ended up asking for him to skip K. They now had a more formal policy (as a result of my DD's experience). We had to do the parent eval., he met with a first grade teacher to do their beginning of the year assessments, and he was given both an intellectual and an achievement assessment. He ended up skipping K. I know that we have a really open-minded and exceptional principal and that we're lucky.
So, my point is, there may be a policy against early admittance, and in some states it can be and is rock solid, but *sometimes* there is a way for them to make a quiet exception to the often-stated "policy." Of course, a law can be a whole different thing. . . One note: here we have a written acceleration plan that requires us to reevaluate the situation along the way and adjust as needed, which might be a good thing to offer a slightly hesitant school district -- they won't be so afraid that they would be stuck with a bad acceleration if it doesn't work.
HTH