If you are giving them one more week, I think you should protect yourself and put it in writing, pointing out the conditions and that they will forfeit the deposit if it does not work, if you are not re-funding.
I have had many AP kids under my care. I wish I could be more picky, but I would not have children to care for if were that choosy, my hours are the shortest, and most people here pick licensed care or nannies and when they run out of options they go to unlicensed daycare.
If you decide to keep her, my advice is to be firm with her and have the same expectations of her as the other children. Kids learnt to differentiate between environments and also learnt when and who to behave well with. I also would not pay any attention to her while she is eating, I would just serve her food and only look at her when she thinks no one is looking. Our job is to provide healthy food, theirs is to decide whether to eat it or not. It is impossible to force someone to eat, so you will loose there every time.
I currently have 2 kids (out of 3) that are AP at some degree. One of them is a 3yo dcg that has been with me since she was 1 yo. She is now awesome! I really going to miss her next year when she has to go to JK. She used to sleep with mom, she was held ALL THE TIME, and cried when mom left the room even in her house. She is the youngest of 3 older brothers and parents are of mature age, so she is the princess of the house, but she has learnt different here. She comes everyday happy and with good attitude.