Yeah, a ratchet is a good analogy.  A simple rear hub off a pushbike and a bike chain would do the trick.  It would freewheel when the engine is driving the kart and engage for reverse when your pulling on the lever.

I like this a lot. Would freewheeling at 40mph be ok for the bearings? The chain, would it be non-continuous, with a spring at one end, drive cog at the other? so you drive back, freewheel forward, drive back, freewheel forward etc, working the lever?

I really fancy this. Seems simple-ish