I also have a 4 year old whos birthday is sep 21, she too will go late to school. I used to stress about it, but not anymore. The way I see it is that I have an extra year to enjoy her, without school pressure of home work. No private school will accept her or put her ahead, public schools go by hiilsborough cut off date. Just enjoy that extra year. I also have an 19 year old son, whos birthday was in nov, I was living in PR and was able to start him in school early, after he turned to high school you knew why they say it is not good to put them ahead. That is when you see your teenage trying to act older, and wanting the same privales that 18 years old have, even though they are 15 or 16. He graduated at 16 from high school, but was still inmature for college. The director from my daughters school told me it was not a good idea either.
But what I can tell you, with my 4 year old, she goes to montessori school, she is already reading, they are not forced to, just the ones who show interest, know their letters and sound, start. So instead of putting my 4 year old in daycare, I selected a montesorri school, so she could have a good foundation, and take advantage of all that she already knew. The montesorri she goes to contnues to 5th grade. I plan to keep her their at least up to first grade, she is too smart and I dont want her to repeat all that she knows in kindergarden again. Try to find a montesorri in your area, you will not regret it.
Good Luck
What I can advise