Switchable current shunt resistor
You could use a small latching relay. Or a MOSFET.
For minimum effect on the reading you'd want a resistance of much less than 250m\$\Omega\$, which is possible with an N-channel MOSFET connected on the other side of R19 (between R19 and ground). For example, an AO3400 MOSFET has less than about 75m\$\Omega\$ when driven with 3.3V. Off leakage is &lt;5uA at 55°C, which may affect your specification for maximum source impedance in voltage mode.
75m\$\Omega\$ implies an effect of about 0.03% and an effect on the tempco of the combination in the 1ppm/°C range.
Careful PCB layout is required in the current loop.
Your low pass filter elements imply a typical low-frequency process control application, but if you are dealing with an atypical situation then the MOSFET output capacitance of ~75pF might be worth considering.