Enforce the law. If it has been duly passed by both Houses of Congress and executed into law by the president ... Enforce it.
I WOULD have a "Deportation Department" ... I'd recruit goons from the Left's Brown Shirts ... they know how to strike terror ... and let it rip.
Due process is guaranteed. However, the steps that immigration "judges" take are not codified or guaranteed by the constitution.
Give them a fair hearing. If it can be proven that they are in the United States of America illegally .. deport them. That is due process.
The line of "bogging down the courts" is a straw man argument, at best.