1. Article IV Section 4, cited by both of you.
2. Article I Section 8: "To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions;"
President Trump can call up the militias of several states to repel the invasion; it doesn't have to be the formal military.
3. Article I Section 10: "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."
States can act unilaterally to call up their own state guards to defend their own borders, citing "imminent Danger as will not admit of delay" to bypass Congress. Since this so-called caravan is planning to enter via Baja California, it is unlikely the Jerry Brown will call up California's guard, but other states bordering Mexico certainly can.
-PJ
The power of Article IV Section 4 is it is a FEDERAL MANDATE (”SHALL”).
Article I Section 8 is more of a provision at the discretion of Congress (rots a ruck).
Article I Section 10 is a good one at the state level - states’ right to take action unilaterally if necessary.