RES showed an internal diagram of a PhotoMOS optocoupler which is basically a miniature solid state relay. Here's a typical one's datasheet:
http://www.ixysic.com/home/pdfs.nsf/www/CPC1017N.pdf/$file/CPC1017N.pdf
Note that its load side is only good for 100mA, and its control side is just like driving a LED - i.e it needs a series resistor.
*IF* the wireless doorbell draws less than 100mA through its switch, the one I linked would do the job. If it draws more current, it would need something beefier.
While a reed relay could also work, they don't like large load currents, especially surge currents while switching, which tends to result in rapid failure. If its more than just a few mA for a logic signal, you'd need a normal relay, and the PIR may not have enough 3.3V output current to drive one.
You cant do it with a single MOSFET - if you use a
NMOS one like your IRLB8743, with the source to the load, you need to drive its gate well above the Vcc its drain is connected to to turn it on and pull the gate right down to 0V to turn it off.
Use a PMOS one, source to Vcc and drain to the load and its more practical, but you then need a driver for it as its off with its gate at Vcc (use a pullup resistor to Vcc) and turns on when the gate is pulled down to a lower voltage. If Vcc>3.3V you cant drive the gate direct from a 3.3V control signal. A low Vgs threshold NMOS one could be used to drive the PMOS gate. See
https://www.eevblog.com/forum/beginners/mosfet-175797/ from last month.