You could purchase gift cards with your credit card, use the gift card to load to a prepaid card like Target REDCard or Bluebird, or Serve, then use the bill pay feature in one of these to pay the daycare. Depending on amount needed gift cards can be bought for fees of $3.95-$5.95 in denominations ranging from $10-$500. At the right places you may get gas points at the market you shop at, or depending on credit card used you may get bonus categories, if say, your card has a 5x bonus at Office Supply stores then you could go to Office Max or Office Depot and purchase them for 5x rewards.
I pay my mortgage in this fashion.