Generally commanders work with soldiers based on the situation, but they don't necessarily have to. Reference AR 600-20, Ch. 5-5:
(1) The Army assists the Soldier in providing for the care of his or her Family members. Mission, readiness, and deployability needs especially affect Active Army (AA), Army National Guard, and USAR single parents and dual military couples with Family members. Plans must be made to ensure Family members are properly and adequately cared for when the Soldier is deployed, on temporary duty (TDY), or otherwise not available due to military requirements. Army National Guard and USAR Soldiers are subject to those policies and regulations, and will implement plans during any period of absence for annual training, regularly scheduled unit training assemblies, emergency mobilization and deployment, or other type of active duty. Emergency-essential civilians who meet the criteria set forth in paragraph 5-5 a are encouraged to have a Family Care Plan that follows the guidelines set forth in this regulation.
(2) DA Form 5305 (Family Care Plan) is the means by which Soldiers provide for the care of their Family members when military duties prevent the Soldier from doing so. It will include proof that guardians and escorts have been thoroughly briefed on the responsibilities they will assume for the sponsor/Soldier and the procedures for accessing military and civilian facilities and services on behalf of the Family members of the sponsor/Soldier. It will attest that the guardian and escort agreed to provide care and have been provided all necessary legal authority and means to do so.
So your commander is within his legal right according to regulation, although I would consider it an abuse of the short term care provider.
I know this doesn't help any, but I hope you can find someone who is willing to pick up/drop off your child for you while you're at work. To provide additional incentive, you can use some of your wife's deployment pay to pay a bit extra for their services, provided you can find a willing participant.
Who do you have listed as your short term care provider? Will fulfilling this role put them in a situation where they could lose their job or would it cause undue hardship?