Choose Single Race, and adjust the settings accordingly :
Nitro : on
damage : off
traffic : off
track : circuits
laps : 1
opponents : 1
vehicles : same
opponents strength : easy
Then pick city/among freaks, and use the Kart.
Nitro boost off the start, but don't bother with any of the chicanes. you can just go straight right up to the end. Do the same thing round the other side of the track, and boost as much as you can.
You'll get the "off track" timer, but will rejoin the track before it runs out. You can complete 1 lap in about 21 seconds, and complete with the save/load screens you will be able to complete a cycle in 50 seconds or so.