> "...but the moon does not have any linear momentum due to rotation"

This is not correct. Why?

[![fig1][1]][1]

At the moment a tractive force is applied through the wheels, and an equal and opposite force is applied on the moon. Newton's 3rd law still applies.

This force accelerates the center of mass of the car _and_ the moon. Both acquire linear momentum.

In fact, total linear momentum is conserved here. 


  [1]: https://i.sstatic.net/AUoop.png