Regarding sample-and-hold chips, I would simply assume that ICs like AD783 with sufficient speed aren't available in your country. Sorry for that.
Sample and hold circuits with OPs and separate analog switches are shown in many OP application circuit collections. I would consider 74HC4066 or other dedicated CMOS analog switches. Compared to 2N7002 they have much lower charge injection, which is a prerequisite to achieve the intended 10 mV accuracy. A MOSFET with substrate diode like 2N7002 is also not suited to process bipolar (positive and negative) pulses. You have to refer to MOSFETs with separate substrate terminal, there are very few types available.