K.L.
My daughter has an end of August birthday and our cut-off is September 30. We also sent her to kindergarten when she had just turned five. She is TINY for her age. I'm pretty sure she was the second smallest kid in her grade last year (out of 120 kids). She is 12 and in the 7th grade. Weighs only 75 pounds. She is smaller than my 10 year old 4th grade daughter. It is what it is!
We had the same conversation that you are now having. It finally came down to the fact that we were going to hold her back a year just because she was small. We chose to keep her going where she was. I have no regrets. She is now in 7th grade and has gotten all A's both first and second quarter this school year. Is she a little immature? (I actually like how you said "less mature" - that makes more sense!) Maybe she is. She sometimes still plays American Girl with her little sister (although she probably would not admit this to her friends). She has friends, but she's not getting invited to parties every weekend. She's a happy kid and I think she's where she belongs.
I don't know if this helps, but I just thought I'd share my story of my "little" girl. Best wishes to you! :)