I imagine illegals would get legal counsel at their deportation hearing. Their defense would be, it's all a mistake? The question is simple: is the person in the country legally? If he is he can prove it. If he can't prove it, he's not.
The Congress and/or the legislatures are charged with making procedural rules. Legislative changes to the rules of criminal procedure for such cases could be made just like all other changes to them have been made.
One of the changes made could be simply that the defendant is tried by a legislative tribunal, much like a traffic court, with a trial de novo application to a court of record tailored to those presumed to be in the country illegally and therefore not having full access to the same protection as a citizen.
In other words, there would have to be some sort of prima facie evidence introduced to indicate a legal status. If so, then the possibility of a jury trial. If not, bye, bye. Yes, I don't see any reason this all can be done under the constitutions, the 14th amendment notwistanding. I think the option of a trial de novo would overcome objections about that amendment.
The original question applies here that has applied all along. Do our elected officials have the will to do it.