That's a great question ! I certainly think you must insist your child gets adequate sun protection. Particularly, that afternoon play time. I have one idea that might help...
Our family has switched to the spray on sunscreen (70 SPF). My son loves to spray it on, because he doesn't have to get sunscreen all over his hands. I like it, and use it everyday...it makes my skin feel good, hydrated. It dries instantly, and takes just a minute to apply. It has proven to give great protection for us. Perhaps the school folks would be more inclined to use the easier spray.
A floppy sun hat sounds awefully cute too..I can just see her!
From an old mom, who has been through the entire school system, sometimes you have to fight a little to get what you want. I think this is one of those times. How can anyone argue that sun screen is not important for young children, and all of us. A school caring for toddlers should guarantee your child is safe, and that includes sun safety. Particularly if your child is fair, and at most risk for sunburn.
You sound like a great mom...I wish you the best