Contrary to how it sounds, you need to use each powerup in a single race, not the entire game...
Similar to The Brian Man's method, but create a game online and just sign in three guest accounts. In online play, the guest accounts won't get kicked out of the game if they are inactive for awhile, so no rubberbands needed. You can use guitars or drum set if you do not have enough regular controllers.
I did races in reverse so that you start on the last race first. This race is the longest and I believe you have a better chance of getting the harder to get powerups. Run at least one of the guest account characters half way through race, and then with your main account player just sit on a group of balloons until you get one of each powerup.
The only powerup that forces you to go forward is the rocket pack so try to use this one last if you can, but the last race is long enough that you shouldn't have any problem with finishing before you get them all.