Is saying "There is no god but Allah" and then responding to it with "Muhammad is the Messenger of Allah" permissible and lawful when two people bid farewell to each other, and what is the evidence for it?

The remembrance mentioned in the question has not been found in the well-known collections of Sunnah, and therefore it is not prescribed. What is prescribed for a Muslim at the time of farewell is to say "Peace be upon you" to the one being bidden farewell, just as it is prescribed for the traveler to say to the resident: "I entrust you to Allah, whose trusts are never lost," and it is prescribed for the resident to say to the traveler: "I entrust Allah with your religion, your trust, and the ends of your deeds."