Battling the "Late Birthday" - My Kid Will Have to Go to Kindergarten Twice

My one and only kiddo was born on September 21.  In many states, including my own, the cut-off to begin elementary school is to be 5 on or before September 1.  Currently, she's in a wonderful childcare situation that offers a private kindergarten.  My husband and I were talking with one of the administrators there and we realized that she will complete kindergarten at the daycare facility -- and then go on to our nearby public school and "repeat" kindergarten again because of her age.  Her daycare will naturally move her into their kindergarten class after she turns five and will spend the majority of the year doing kindergarten level work.  Once she "graduates" from the childcare, she spend the summer at home and then go to our nearby public elementary.

I'm feeling a bit frustrated that she'll be turning six just a couple of weeks after the elementary school year starts -- and worried that having her "repeat"  material in kindergarten will lead to her feeling frustrated with "big kid" school.  My child is not a genius by any means, but she's doing nicely in preschool, picks up material quickly and socializes well with others.

Has anyone run up against this issue?  Did you go to your school district office to learn how about kids "test out" or "up" from a kindergarten into 1st grade?  Or, did you visit private schools to see if they would be willing to take your child with a late birthday?  OR, and maybe the biggest question of all -- did you just send your child to kindergarten at 6 and find that everything turned out okay?  I would love to hear from other parents who are in similar situations.