S.K.
I would get the daycare to give her lunch early. Maybe she just needs her nap earlier than the other kids. All kids are different in their sleep needs. I know my daughter used to want her nap at 11:00 and took a long time to get her to take it later than that. She could stay awake a long time in the afternoon but the morning times she just was sleepier.
I would definitely want her to eat lunch. Snacks are good but I would hate to get out of the habit of eating lunch. That's just my opinion. Hope it helps.