As I read this and OP's new posts I am more confused by OP's account.
It appears that this child's mom is afraid of a DNA test for some reason, thus the acknowledgement of paternity was signed and it is very possible that there has been some judicial order, whether or not OP knows about it or not. The are points where she says mom and baby have always lived with OP and others where she states mom and child lived with the child's father. Sometimes this "Living together" is also termed visitation.
The story about what happened at the hospital is very distorted. But Lidj's remarks about what she has seen happen elswhere does nothing to clatify what happened or the implications of the facts. I have a feeling that much of this coercion is in OP's mind, based in part on her expectations for where her daughter and grandchild will reside. Since OP's daughter and grandchild also resided with the legal father after signing the Acknowledgement of Paternity, it is very likely that dad will get some custody if he doesn't already, visitation, and the child may have to remain in Kansas. While the father may have a charge against him for contributing to the delinquency of a minor, has he been convicted? An arrest or conviction for check isn't likely going to make much of a difference re paternal grandmother being a caregiver, especially if that is because that placement was necessary due to the DV RO in place.
With unmarried parents, where the father is asking to establish paternity and willing to sign the Acknowledgment of Paternity that is encouraged, this is due to an act followed by most states and has nothing to do with coercion or the fact that you live in a rural area. It is very possible that it was explained that if the couple didn't sign these forms, then the court would order a DNA test, this may be the test oP's daughter didn't want, this can but is not always a blood test, it can be a swab. The hopsital worker checking out the carseat as the child leaves the hospital is a common practice because studies have shown that most are not properly installed, it is an educational process and yes a hospital may refuse to release the child if the parents refuse to allow the inspection. It doesn't matter who installed the restraints or how many years of experience they have, refusing the inspection will result in delays. It really sounds as if much of these difficulities are caused by the actions of OP and her daughter, possibly through ignorance.
It sounds as if daughter was arrested for violating a domestic violence restraining order. And while their argument may have began about the baby, apparently the baby wasn't present so she wasn't defending her baby and mom didn't report the "kidnapping" of her child to the proper authorities. If she had custody and the father had none, she should have been able to ask the police to assist in the return of the child, but I don't think the account we are getting is accurate or in chronologically accurate order. It is possible that there was some domestic violence prior to this event and dad has a DV RO before he took the baby and that is why the police didn't force the return of the baby? Because the baby was on the order? Which means a "judge" signed some sort of an order, whether or not anyone was friends or related to the police.
The fact that the daughter has bruises on her body doesn't prove she is a victim. The information about the carseat, formula and clothes for the child, theft? Please, these items were intended for the use and protection of the child, no matter who originally bought them. Dad had access to them, you would rather the child be unsafely transported without them, or to not have a change of clothes or food to eat until more could be obtained? This is evidence of dad looking out for the best welfare of the child. If anything, this is more evidence of the removal of a child during a domestic dispute more so than a kidnapping which is usually planned out thus only the child is taken.
OP and daughter need to sit down and get their story straight. OP's daughter needs two types of attorneys now, one for criminal defense or a public defender and a family law attorney. Much of this could have been avoided if custody and visitaiton was appropriately established rather than trying to avoid it in order to remove the child from the state and from the father.