My solution:
Border is open to anyone who wants to apply for refuge or entry. Put the military on the border to shuttle migrants to staging areas where they can fill out paperwork. Use government workers who are nice and slow and can process one person every couple of hours. The rest will wear out and leave after a few weeks. Those that pass, get in. Those that don’t pass don’t get in. Business as usual, just slowly.
Guantanamo Bay Detention Camp? After filling out paperwork, put them in camps in Puerto Rico under guard. Let them stew there for a while, then dump them back into Mexico.