My daughter‘s birthday is sept. 26th. 5 days before the cutoff. Wife and I went round and round on her starting kindergarten at 4 years old. I wanted to hold her back a year. To my chagrin, she started at 4. TBF, she was bored as hell at the daycare, so the wife won. She had some struggles here and there academically, but nothing major. She is now in 7th grade and thriving in all areas except for sports. She is easily the shortest in her class. I still wish we would have held her back and the wife still thinks I am crazy.
I guess if I were in your shoes, and my child was struggling socially, academically and athletically, it would be a no brainer for me. Do it now as if you wait another year or 2, it will be too late.