Strictly as an academic exercise:
Article I, Section 10, under 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."
I believe some have argued this as a basis.
Also, Article 4, Section 7 of the Texas Consitution: Sec. 7. GOVERNOR AS COMMANDER‑IN‑CHIEF OF MILITARY FORCES. Heshall be Commander‑in‑Chief of the military forces of the State, except when theyare called into actual service of the United States. He shall have power to call forth the militia to execute the laws of the State, to suppress insurrections, and to repel invasions.
On the other hand, where in the Constitution does it give the US the authority to control borders. Commerce regulation is permitted, rules of Naturalization is permitted, but not actual physical control of the border.
None of this means that what Abbott is doing is reasonable, but when there are arguments to be had that there are things that Abbott can do.