What is the purpose of a power MOSFET in this voltage-peak detector?
The MOSFET is used as reset circuit. Why BJT isn't used? Because it would interfere with the sample/reset circuit, as the MOSFET has a insulated gate and low Rdson when triggered - empty the capacitor aka. reset/clear.
However you do need just a very small MOSFET for that purpose.
EDIT:
The bottom circuit seems to be done by someone who has skills in electronics and he knew what he was doing, the upper circuit is made from someone who doesn't really know what he was doing, me neither. Maybe it is useless, so you should rethink on what you need from this circuit.
EDIT 2:
In these days, IMO there are such opamp that could amplify and remove bias even with a single supply.

If this holds true, then:



simulate this circuit &ndash; Schematic created using CircuitLab



simulate this circuit