Don't let in people without a device and ban people who are caught stealing once. If they force the doors and come in call the cops and prosecute for felony burglary even if they were only originally banned for stealing socks originally.
Share bad lists and stealing socks means that you are suddenly banned from every store in 100 miles.
Make exceptions for curbside pickup and pre scheduled trips to the grocery store/pharmacy where you will pay for the security that will walk around with you.
Expire people off the list after 5 years of one theft or never after a string of thefts.
For the normal customer nothing happens your phone silently attests from your pocket that you aren't on the list by pinging a service and retrieving a token not sharing your info with the store.
Losers here an alarm sound and a voice telling them to buzz off on penalty of jail.