Hmmm... 🤔
"states have a right of self-defense, under Article 4, Section 4 and Article 1, Section 10, Clause 3 of the U.S. Constitution."
https://constitutioncenter.org/the-constitution/full-text
Article 4, Section 4:
"The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened) against domestic Violence."
I don't see a) anything there that defines "invasion" or b) grants the states the power to act if the United States chooses not to.
Article 1, Section 10, Clause 3:
"Section 10: Powers Denied to the States
No State shall, without the Consent of Congress, lay any Duty of Tonnage, keep Troops, or Ships of War in time of Peace, enter into any Agreement or Compact with another State, or with a foreign Power, or engage in War, unless actually invaded, or in such imminent Danger as will not admit of delay."
Again, "invaded".
It's pretty clear from section 10, since it's also speaking of troops, ships of war, and engaging in war that it means MILITARY invasion, not an influx of citizen refugees.
Dictionaries at the time back up that reading:
https://mises.org/wire/what-did-founders-mean-invasion