Hi,
the idea to do this with a MOSFET is fine. Some things need to be considered, however:
- As your input voltage is possibly lower than a MOSFETs turn-on voltage, you need an external supply voltage for the switch anyway.
- What you want is a low voltage drop, so a MOSFET with a low RdsON.
- The simplest thing here is a fat N-Channel (less RdsON than P-Channel of equal size) MOSFET like an IRFZ44 in the positive supply rail. The gate voltage can be drawn from a 9V block battery with a switch.
A Zener protection for the gate and a resistor can be used for protection. An IRFZ44 can handle several 10 Amps, so your 2A is no challenge and the voltage drop will be low.
- Battery lifetime is very long because the MOSFET gate draws no current.