If we sent our military 10 miles into Mexico to set up a border zone...
Mexico has (or at least had) such border zone. Except for Baja California, Mexico has a Frontier Zone along its border with the U.S. If you drive more than 20 km south of the border you will be stopped at a checkpoint and a have to get a visa allowing you and your car to travel further into Mexico. So the Mexicans have already established the checkpoints for our military’s use.
But my point is, if we did something WITHOUT their permission, against their will... Ellison and his ilk would respect their wishes. Just not ours.