The Golden Weasel Award is one of the four awards that can be received in Time Trials mode. This one is incredibly hard to receive, especially in the last races. This award's concept resembles that of the Author Time in racing games, e.g. in Trackmania there is Bronze, Silver, Gold and the Author Medal.
Due to strict times, all you are required to do is to master your skill. Try learning and memorizing the track and fastest path possible. Also, be sure not to rush, and be calm, patient, agile, smart, and this Weasel Award is achievable.