Perhaps by virtue of them disobeying the basic laws of this country, before, during and after their border crossing, makes them invading horde, and not simply "undocumented" immigrants.
I believe that is the whole push behind the PC terminology change, as an effort to make the 14th read more favorable.
First you would need a court that would even take the case. This one almost certainly would not.
Before it could even get to SCOTUS the case would have to wind its way through the circuit then appeals courts, and then you’d have to hope the Supreme Court would even give it a hearing (the lower courts would already have likely ruled against any challenge to current interpretation of the 14th).
This is why I get back to the idea we need 4-5 new Supreme Court justices. What we need is a court that would be willing to either take the case on appeal or pluck it out from the lower courts. We’d need a SCOTUS that was interested in hearing the case, preferably one that would encourage it to be expedited through the system.
If we had a decent bench of justices you could probably achieve this if both the Legislative and Executive branches forced the issue. But all this relies on a different SCOTUS and a Legislature and Executive that challenge the status quo. This is not going to be an easy thing to achieve. There would really probably need to be a sea change in attitudes on this issue by the vast majority of Americans to nudge the courts to even listen.