Crossed Wires
As the A320 would be operating under an AOC, the necessary ac paperwork would be carried in a folder kept on the flight deck.
Your pilot's licence which lives in your flight bag does not require a MCC certificate to be carried.
As you say, the MP type rating implies that a MCC certificate was issued, if the rating was issued on or after 1 July 1999.
Prior to 1 July 1999 (JAR) no MCC was necessary for ac over 5700kgs.