he can get the phone number and address based off where you live and the little info you've already given him, but if you already gave him the name why not give him the number and address?? unless she is not a licensed daycare how can he use it against you?? i would throw a hissy fit to not be provided with my childs school/care provider's info.
unless he's morally unfit, why do you feel the need to hid it, if he's not on the pick up list, he can't just walk in one day and pick her up, unless he brings joint custody proof (which is unlikely) and even still the day care if reputable would probably still notify you before allowing him to take her. i know it's scary but just like marrying a man with children is a package deal, getting a divorce, this is a package deal, and if you refuse the info without probable cause and proof of that probable cause, judge aint gonna be happy with you
something i would STRONGLY advise you, call your attorney, give him all the info this man may want, dr, school, daycare, etc, EVERYTHING and let your attorney know you're going to start having him call lawyer instead.
this way you are not in the wrong for denying info he has the right to have and there are no conflicts between you two that can hurt/help your case, and unless he calls to talk to the child, you SHOULD tell him, you need to call my attorney and leave it at that. i wish like hell i did that, and instead talked to him myself, and lost most my case and took me 8 years to really build one up