... Should the transistors operate in saturation region?( my MOS are in cutoff at the moment)
The first branch senses if there is current flowing. The only devices that should not be off are these two devices. And you should make sure that the output of the inverter is at the correct value.
How exactly will this LDO/charge pump be used?
If you are using the bandgap to provide a reference voltage for an error amplifier for another input signal that you want to use for your sensors, I can show you a relatively simple topology that might work for that.
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