Definitely Postman otherwise it would be Mailman Mike instead of Postman Pat
Except here we can no longer officially use the word "man" in anything (even when the person is male), because it is not politically correct. So it would be "postal carrier", "letter carrier", or "mail carrier".
police officer
fire fighter
chair person
sports person
Since I don't like PC, I will continue to put "man" and "woman" at the end when referring to specific people
Although I do often use the neuter version when referring to the generic/collective. Some really sound beyond strange though-
fisherwoman? fisher person?? Maybe just "fisher"
Where I work, when we had a female in charge of the Board, people kept calling her the "chairman" and I refused to do that. I called her the "chairwoman" since it seemed really odd to say just "chair", like I was calling a person a piece of furniture