Is this a bug, or is there some setting that will allow me to transfer the contents?
No, this is not a bug. Child nodes of a group node are not considered neighbors of the group (unless there is an edge that connects each child to its parent group). For this reason, childen are usually not displayed in the neighborhood view.
You could try collapsing your activity before selecting it. After converting the neighborhood view with the collapsed activity into a new document, you can open the collapsed activity in the new document.
Note, this approach does have side effects if one of your activity's children is connected to a node outside the activity. In this case, collapsing the activity will reassign the edge from the the child to the collapsed activity. And consequently, the node outside becomes a neighbor of the collapsed activity.
Selecting the children in addition to the activity does not work because the neighborhood view does not include edges connecting the selected elements.