what you plan to do is a logical step in realizing a low noise In-amp. For the switches, there are certainly plenty of options. I will go through them below:
Assuming you are using a +/-12 V supply (or similar), you also want your switches to be able to handle such common mode voltages.
CMOS analog MUXes
The simplest option is probably CMOS analog switches. E.g. the DG409 is an industry standard 4:2 MUX available from many manufacturers that can handle +/-15 V. Btw it is better to use switches on both sides of \$R_G\$, which is why I recommend using 4:2 MUXes. The reason is the switch capacitance slightly attenuates the gain of the corresponding input op-amp at higher frequencies and that will look like a fake differential signal after the in-amp. If you add similar switches to both nodes, both op-amps will experience a similar high frequency attenuation granting slightly better CMRR.
Better yet is to divide \$R_G\$ in two halfs and place the switch in between. If you want low-noise you are probably going to use rather low feedback resistances. So the switch on-resistance should be <10 Ohm. Many such parts are available, e.g. from Analog, Maxim, TI, Vishay etc. Low R switches also have higher C typically because they are MOSFETs internally.
One downside of CMOS switches is that they are referenced to the supply rails. Therefore, when the signal voltage changes, the switch on-resistance also changes. If you want to built the in-amp like you plan to do, the switches carry the full current flowing between the two input buffers. A changing switch resistance will therefore change the gain depending on the signal level. This is also known as harmonic distortion.
There are a plethora of optically isolated MOSFET switches as you suggested, the ones you are looking for a usually called solid-state relays with MOSFET output. Make sure they have this kind of output configuration:
The output side floats on the signal level, eliminating the dependence of the on-resistance on the signal level. These switches also have off-capacitance, so the same argument about using two switches instead of one applies.
As for CMOS switches, on-resistance can be very low (even well below 1 Ohm) at the cost of higher off-capacitance. Typical suppliers would be e.g. Toshiba, Panasonic, Omron, Vishay, IXYS etc.
If you need very low on-resistance and very low off-capacitance, mechanical relays are great but also quite complex to implement. You want sealed ones if your want to handle signals reliably long-term.
Better in-amp topology
The in-amp topology that you want to build is ideal for ICs because gain can be set using one resistor. But it is a bit worse if you want various gains, because your switches have to carry all the current flowing between the two input op-amps. Building a discrete in-amp allows you to avoid that issue and select switches with lower capacitance and lower current rating by doing the following:
simulate this circuit – Schematic created using CircuitLab
Another advantage is that you get to choose the input op-amps according to your needs. A disadvantage is that you have to also build the difference amp stage which requires extremely well matched resistors for good CMRR. But one can of course use an integrated difference amp here, or even an integrated in-amp with fixed gain.