I would take three deep breaths, stand up straight, strike a power pose for confidence and then walk in there and demand (politely) to have your child moved up. There is no up-side to delaying any longer. Don't worry what people think of you. It's sort of lame imo to be leaving it up to a 4 year old. You decide where she goes to school and what class she's in. She will adjust. Most schools do not make such a big deal about sensitive kids. Go for it!
My daughter got to go to a private K4 with lots of school prep. She started at your daughter's age. It was such a HUGE boost for her in every way. I moved her there intentionally to get her out of a more "daycare" play environment. Best move ever. This won't be the first big adjustment in your daughter's life. She can handle it and so can the school.
Also, that age is crucial for discipline. I welcomed the opportunit to discipline my daughter if she misbehaved early on in school. She didn't, because we had already disciplined her at home and other little toddler classes to be able to participate without disrupting. Do your part to insure she follows rules. If the teachers also do theirs, there should be no problems. My daughter is now 7 and never needs discipline in any class settings. We homeschool and attend lots of lengthy lectures, concerts, tours where good behavior is needed, she has music lessons and recitals and sits through concerts, she was awarded student of the year in Tae Kwon Do....this is all a breeze, but the foundation was ages 2-3-4-she needed discipline then to insure her maturity now.