The OP’s daughter is in fourth grade. In my opinion, this child has no clue as to what boarding school really entails. And that’s good. My advice to the OP is to casually and gradually educate yourself about boarding school, but don’t involve your daughter until much, much later in the process. Let her be a 9 year old! (Or 10).
Also learn about the many, many excellent schools besides Lawrenceville.