Reverse current protection for a battery-operated circuit
You only need a reverse polarity protection for your device if there is a possibility that the power input can get applied in the wrong direction. You would have to ask yourself how likely it would be that a reverse polarity condition could happen and then decide from that whether you need protection for that scenario. 

Most simple reverse polarity protection schemes are going to involve the use of a diode in the input power path. You can select a Schottky diode to minimize the forward voltage drop so that you can get more usable voltage range from your battery supply. There are diodes available that have pretty low forward voltage. For example a generic SB30 power diode would come in at a Vf of ~400mV @ 100mA. 

If you do not mind getting a bit more complicated you can also use a discrete PMOS FET to provide reverse polarity protection like shown here. One recommended P-FET for this circuit could be the Vishay Si2323 which would show only .068 ohm resistance when the battery voltage was at 1.8V or more. FET selection needs to be done carefully to find one that can be fully on at the minimum battery voltage. At the 1.8V this FET can easily support the 500mA rating you called out for your LDO regulator.



The MCP1825 is not going to provide reverse polarity protection for your device. If you look at the functional block diagrams in the data sheet you will see that there is a diode inside across the internal PMOS FET that would forward bias in a reverse polarity situation and allow a negative bias to be applied to any down wind circuitry.