I wouldn't try it. In our district, if they find out later they are known to "demote" the child. We had one go 5 months in K only to be pulled out for false documents. In this case, she had falsified an electric bill to get into a better school. I can only imagine the consequences would be the same or worse if you falsified a birth certificate.

It would be better to find out your district's policy on early entrance and work with them.