This can be done in local multiplayer
When choosing your game mode, before selecting it with A, press over on the D-Pad to make it two player. Any track, any number of laps, it doesn't matter. Just set the number of AI to zero.
In the Cops & Robbers mode, there are lots of obstacles placed all over the track, such as boxes, cones, signs, etc. Just start driving around the track slowly enough that you can't miss them. You won't have any competition.