>> How could you accomplish this without deporting them?
“Encourage” them to self-deport.
Self deportation is the ticket.
Shut off the spigot of money, free housing, free health care and sweeten it with a small sum for biometrics surrender at the border and a signed agreement not to return for 10 years and then only through legal channels, punishable by prison time. Negotiate a fee for Mexico or their own Country to imprison them if they come back to USA illegally.