What you're describing is a civil law enforcement mission, NOT a military mission.
And what follows after that is that the mission will get progressively less clear as time goes on. There will be a few "unfortunate incidents" early on, where a US citizen gets shot because of unclear ROEs. The ROEs will be "clarified," which will in turn make the original mission less clear. (You have to have been in the Big Green Machine to understand how this works.)
Eventually, you will have soldiers being offered bribes--and more than a few taking them. At that point, you might as well write off the USA, because once the military goes bad, it's a hop, skip, and a jump to the final collapse.
During WWII when we were at war, we had troops on the border and on major bridges, etc., within this country. This just makes sense. This is an international border, and our borders are the first line of defense - so it is very much the province of the military and I believe they can and would do the job given them if our leaders would get a little backbone and do it.
Our President could greatly reduce the need for the military if he would only start speaking out against this and stop encouraging their coming. Let's take the laws we have and enforce them in each and every way possible.
There was a time we had a border - does anyone remember that?