How set in stone is the K cut-off where you live? I ask b/c, in my state, each district can set their own cut-off although none of them are later than 10/1 b/c that is the latest date the state will fund. Our governor did sign a bill allowing for funding of early entrants a while back however, if the child is a "highly advanced gifted" kid. I don't know what type of testing they do to ascertain that, however.
We have two girls with bds right around that same time & a district with a 9/15 cut-off. My oldest just made it and my youngest missed the cut by two weeks. The youngest we managed to get around the cut-off by starting her in a different district with a 10/1 cut-off and then moving her to the one with the 9/15 cut off in 1st. They generally don't make kids who have done a year of public school repeat the grade. We were told that kids who had done a year of private K, on the other hand, would likely be forced to enroll at K students again.
Our other option, had the one district in which we started our youngest not changed their cut to 10/1 the year dd started, was to do both K and 1st in private school and then the public schools told me that they'd let her continue into 2nd without repeating.
I'd look into whether you have any 'sneak around the cut-off' options like that.
