I was thinking how would a chain cope instead of the drop gears? This could remove the idler and get the gearbox to run in reverse while the engine still running forwards.
Chains have been used in many engines to transfer power from Crank to gear box. The Saab 2.2 engine being one. It used a triplex chain IIRC. Back in the 80s there was talk of a chap in Greenfield that had done a chain drive to replace the Mini Drops. Never found him or any other info. Not sure it was a Mini engine though.
A Saab 2.2 engine in a Mini won the "Trafficlights GP" every time, (depending on wheel-spin).
I found that it had a very strong desire to continue straight on, at corners.