I would start getting her used to putting herself to sleep. Okay, so you now know that you can't rock a day care kid to sleep but you did commit to not letting her CIO.
Focus on getting her settled before your other places fill whilst you have time to address.
Lay her down awake. When the cry, rub back a bit and leave. Let cry for 10 mins and then go back, lay down, rub back a bit and leave again. No other interact. No chatting. No picking up. Will take a few days because it's not as quick to teach as CIO but at least it mean you not break your word.
For future requests - consider if you can do for all children. Can't rub 4 backs at same time, can't rock 4 kids at same time, can't carry 4 kids at same time. Ask yourself can you do this request for all at same time. If answer is no, then it not a reasonable request from day home.