Have the automated system that categorizes the calls tell people "your call number is $num, please record this number, press 1 to continue, press 2 to hear this

Have the automated system that categorizes the calls tell people "your call number is $num, please record this number, press 1 to continue, press 2 to hear this message again" The logging system should already track the call that way, exposing some unique ID (or some hash of it) to the customer for record keeping purposes is unlikely to require major engineering effort.