The "option" may not be viable, but if we don't get them out the republic isn't "viable" either. "Rounding them up" may not work, but there is a way to get them out in a heartbeat.
The Constitutional option is to call up and deputize the national militia, by definition every adult male between the ages of 18-45. Put up a $5,000 bounty on any illegal within US borders with a minimum $25,000 penalty for false arrest and imprisonment. Put up a web page on how to make a legal arrest with references to all the appropriate case law. Enforce strict penalties for anyone harassing or detaining anyone here legally.
There would be a traffic jam at the border, for months.
Thought of you when I saw this story.
http://www.nbcnews.com/news/us-news/bounty-hunters-mistakenly-bombard-phoenix-police-chief-n405341
You may want to keep track of what happens next so as to bolster your position.