M.H.
You need to carry on doing what you are already doing. If you're ok with her coming over occassionally maybe you could talk to her parent/s and give them set times that are convenient to you for her to come play. Do not feel obliged to give her snacks, if she says she is hungry send her home, she only lives next door. Maybe if the snacks stop coming she'll want to come over less anyway. It is a bit strange that she wants to come play with a four year old but it sounds to me that maybe she gets more attention at yours and that is why she keeps coming so frequently. The idea of putting the sign on the door is a good one but you need to explain it to her that she may only ring the doorbell when it says David can play, she needs to go home if it says he cannot.
If you're really not happy about the situation you'll have to tell her mom not to let her come over anymore, it is too disruptive.