I think I understand your dilemma. We can't very well claim we respect a parent's right to instruct their child as they see fit and then undermine them when their views don't mesh with ours. It doesn't seem right.
However, surely the parents know what you believe and that you don't keep it hidden. Surely they know that if they let you take their son to church he will hear biblical teaching there. As long as you aren't being sneaky about it, I think you can continue to be the person you are and do the things you would do with any grandson, including read Bible stories to him. If your daughter doesn't want you to do this, it seems to me that it's on her to ask you to stop. At that point you may have another sort of decision to make. But otherwise, keep acting like the believer you are.
I teach Bible lessons to a lot of kids in Sunday school classes and at Christian camps. A surprising number of them come from unbelieving homes and sometimes even have parents who are actively hostile toward Christianity. Yet for some reason, those parents still send their kids to church or camp at let me teach them. Okay, then I teach them. I'm not being sneaky about it; if the parents ask, I'll tell them exactly what I'm teaching their kid. Those parents are usually after free child care, or want to allow their child to spend time with a friend, and probably think they can easily counteract anything the kid might be taught. Well, maybe they can. But maybe they figure wrong and the gospel will grip that child. The parents decided to entrust their child to me knowing who I am. I'm not being underhanded; I'm just being myself.
I do take pains to avoid directly criticizing a parent to their child. For example, if a child tells me their dad says it's silly to believe Jesus rose from the dead, I'm likely to respond with something like, "Yes, some people think that, but the Bible says..." I will NOT say, "Well, your dad is the silly one if he says that." I think parents do expect that if they entrust their children to us, we will speak respectfully about the parents. To fail to do this is, I think, a breach of trust and a failure to act with integrity. So I remain as respectful as I can be even when, sometimes, I have to openly disagree with something a parent has taught a child.