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i´d use the second one with a voltage ref at the pot input.
And a capacitor at V+ input to avoid noise - especially during the pot adjust.
With this solution you always need a current meter to adjust. If you need some fixed values like: 10mV, 20mV, 50mV... i recommend to use a rotary switch to select them.
Maybe an advantage.
if you have a power supply with 5V for example you could use a 40:1 voltage divider at opamp output. As feed back use the output of the voltage divider.
Benefit that i expect is less noise, lower zero level (maybe 1mV). and additionally you could use a comparator with 4.5V to OPAMP output to detect overload.
It all depends on the used opamp.
I would suggest to use a precision voltage reference after the power supply and then feed the output from this to circuit in option -2 (with opamp buffer). As suggested in previous posts, capacitors will help suppress the noise which will can be nuisance in this voltage range.
The datasheet for an LM317 shows that it needs input and output capacitors and a load current of at least 10mA. Since yours has an output fixed at 1.25V then its load resistor should be 1.25V/10mA= 120 ohms or the output voltage might rise with some LM317 ICs.
Your opamps are not powered so they will not work. Since their input voltage is close to ground then some opamps will need a dual-polarity power supply.