To start I realize I don't know your daughter. I've been through this twice (both DDs started private K at 4yo). The first time I chose to enter DD1 into the public school in an age appropriate 1st grade after doing K,1 at a private (mainly due to social concerns that you stated). She learned school was social hour. I had to re-advocated for acceleration in Math which caused scheduling difficulties. Yay for Middle School.
DD2 was in the same situation but with a later birthday. We chose the opposite route but after 2nd grade. This is her first year in 3rd at public school and she's doing great. putting her back to 2nd would be riduculous. (And I don't think my kids are as ahead as some of the others on here).
Lessons learned: don't over-estimate the "rigor" of 1st grade. I didn't see much. Do take an honest look at many of the posts here about what is taught is various grades maybe under "age-appropriate". Go look at some on-line curricula sites or state criteria for each grade and see if your private K matches up. I agree with asking the teachers at the Private school. maybe even talk to parents that have made the switch.
FWIW (to help w a baseline for your decision): our private school taught out of texts that were a year ahead of grade level already (so my kids were actually doing work 2 yrs ahead w no issues). 1st grade math at public school in many if not most places won't include multiplication, addition or subtraction with regrouping. DD1s first grade class at public, they sent home very simple books-a little more than seuss but still very big print and not many words per page. It got kind of ridiculous. I think my kids were doing many AR tests in private school on Seuss and Clifford type books in Kinder (at 4) and moved on to chapter books in 1st (at 5).
OTOH - a friend chose to hold her son back with an August bday even though he was quite bright because he was competitive and didn't work as fast. He had done Kinder at the small church class. She was worried he would get super upset if he wasn't the first done, etc. He's really bright and while he's fine and it was probably a good decision for him socially, I don't know that his school work is especially challenging. He has missed the GT cutoff in our district but he certainly fits the profile otherwise and I suspect he doesn't test well on the group tests. He isn't unhappy though and neither is my my DD1 but they aren't super challenged.
Good luck whatever you decide. I also want to add I think kids will adapt to the large/small group. My DH was worried about this and it was not a huge deal with my kids or my friends' kids.
