B.W.
You should just worry what is best for you son. Do not worry about MIL and how she feels. If you feel he will be better off at the other babysittter, then by all means make the other babysitter the primary. But if you want to "keep the peace" then suggest that on certain days MIL babysits and the other days the other babysits. Does the backup have other kids? If so, I think it is important that he interacts with other children also.
I hope this helps.