power supply with Battery Backup -MOSFET reverse polarity does not work as intended
Yes, you can do this with a MOSFET.  It is done by operating the MOSFET in the third quadrant.  Basically the MOSFET is run backwards so the body diode conducts when the circuit is powered.  Once current is flowing the MOSFET channel conducts reducing the voltage drop across the body diode.  You'll find example circuits if you search MOSFET third quadrant operation.