Except that South Dakota's National Guard has no legal authority or presence on the Mexico border?
Yes they do it's the UNITED STATES of America, when a enemy attacks one of the states they attack all the states
People from one state can join under the supervision of another state, but can not operate independently in another state.
The only that can happen is if the federal gov calls them up and they act under federal jurisdiction.
When it comes to inherent rights like defense, you can hire anyone you want to help you defend your inherent rights.
But immigration law is not an individual right but a federal regulatory statute, so then ordinary people, like a state national guard, can not at all enforce those regulations. Only the federal government can.