It really depends on your level of patience in my opinion. From what I gather you are having a pretty rough time with the child if you are on here seeking advice lol

If you are the type who "works with " the parents you could come up with a plan of action like you mentioned so that no one pays attention to the whining and it may help to stop the behavior.
But if your patience is wearing thin,you are not at your best for him OR the others in your care and that's not fair to you or anyone else.

Personally, I would terminate. When it gets to the point where it is affecting everyone;s day, it;s not good.
If the mother is not implicating herself that would also rub me the wrong way.
In short, do what you feel is right !
Good luck and keep us posted k?