Yes, if I understand you, the Child and Dependent Care credit and the two Child Tax credits are independent of each other, so you are eligible for both.
As an aside, do you have more than one child dependent? A quirk of the 2441 is that you can allow $3,000 for one kid or $6,000 for two or more kids, even if you are not paying for child care for the second kid. As the Instructions for the 2441 say, "The maximum amount of work-related expenses you can take into account for purposes of the credit is $6,000 if you have two or more qualifying persons even if you only incurred expenses for just one of them".
BTW, NCperson is correct that the $3,000 is on the net of what you paid, so you can ignore the effect of the 10% discount.
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