Unfortunately, it is currently not possible to change the behavior of the predecessor and successor context views in this regard.
However, you can try to use "Tools" -> "Select Elements" as explained in How to unlimit neighbourhood / predecessos / successos to select the desired set of nodes. After selecting the desired set of nodes, use "Edit" -> "Copy" or CTRL+C to copy those elements and paste them into a new editor. When doing so the edges of the induced subgraph will be copied as well.
Finally, please consider voting for Better neighborhood/successors/predecessors view.