For AZ to exercise their right to protect its citizens is must organize their own Militia like the Texas Rangers. They have this right according to the Constitution. Then use the Militia to patrol near the border so as not to interfere with the Border Patrol. The Militia should have all authority to even shoot direct threats. The Militia must have the right to destroy gangs and drug cartels within their border. The Militia should have the protection of the Governor and their Justice Department.
I do think there are a number of other steps Arizona might take.
The Governor might declare that Arizona is being subjected to a purposeful foreign invasion... which is true... and send a message to the President that declares that invasion is an state and national security emergency. Arizona should demand, using the tools that the Constitution provides, that the Federal government protect its borders...
I do believe that States have both the right and the responsibility to defend themselves from foreign invasion... which is not a right which is pre-empted by any Federal failures to provide for a proper common defense...