Well, if you have a modded Nintendo Wii and an SD card/ USB, I can suggest a method using GCMM (https://github.com/suloku/gcmm). Otherwise, you're left to unlock this one mini-game yourself by playing Party Mode on one save file, which can be tedious. If you decide on the latter course, Toy Dream has 9 blue spaces and 1 happening in the first 10 spaces, so high chance of a 4 player mini-game. If you play a 4 player mini-game other than the desired mini-game, you have to reset.

There are numerous categories within mini-game mode, which one are you referring to? I'm afraid that there is no way to "disable" a mini-game after unlocking it. You will have to do a new save file with only this mini-game unlocked. Are you playing on console or emulator?