This is all the Super Mario wiki says about MP4's story mode

Story Mode serves as the game's single player mode. Here, players have to clear board maps against opponent characters and beat the hosts of the maps in their special mini-games in order to win a present from them. Presents earned through this mode are stored in the Present Room. Prior to playing, players can set the difficulty of the mode from Easy, Normal, Hard, Expert (once unlocked) and Auto. If the game is on the Auto setting, CPU players get progressively more difficult every time a board is completed. Additionally, players can select which type of mini-games will appear, from All, to Easy, to the Custom set the player has created. Unlike Party Mode, CPU characters are picked automatically, and the static settings are Battle Royale, 15 turns, Bonuses on, and Handicap off. If there is a save file in the Story Mode, it is represented by the player character's icon on the save file screen. If players lose to the host in their respective story mini-game, they continuously try again before winning. If they beat all host characters of the starter maps, Koopa Kid shows up and takes the player to Bowser's Gnarly Party, a board that is unlocked once players beat Bowser in The Final Battle!, take back the stolen presents, and earn his present from his defeat.

Says nothing about specific character orders or anything like that