If the new school won't deviate and will make him repeat K, perhaps it isn't the best fit for him either. Do you have any other options other than his current school and the new one that won't consider letting him continue on as a first grader next year? Is the fit problem at his current school related to the school as a whole or the teacher/classroom this year?
My kids are at that young for grade end as well and we've found that a lot of local districts will make a child who was too young for public K and who already did private K repeat that grade. However, if the child does private K and 1st s/he is allowed to continue on to 2nd without repeating a grade. I don't know if that makes sense! Tell me if not.
FWIW, my youngest has a late Sept. bd and is also very small for age. This year, at age 9, she is in the 10-15th percentile for the first time since she was under one. She has been in the 5th percentile for years. I don't believe in holding children back grades when they are otherwise ready solely b/c they are small. With her genetics, my younger dd is likely to be 5'1" at best. She is eventually going to be among the shortest again once everyone finishes growing.