If they are leaving anyways then so what if the mom gets upset. Yes I get her point the child has been through a lot and moving him probably isn't a good idea.
What about using the argument that moving to the other provider for the last part of the school year - even if you say following the long weekend in May so another couple weeks to give her a chance to try out the program and explain it that way to the child as well. If he likes it you will try to make plans for him to go back there in Sept and if not then have the summer to make different arrangements.
Letting her know that as of the end of the long weekend you will no longer be doing the school runs so she needs to move by then or make her own alternate arrangements which would then be her responsibility to contact the other woman and arrange the drop-offs not yours and her responsibility to pay for the service not coming out of your pay.