It doesn't feel right. Adding some inaccuracy into a circuit, and trying to fix that with a precision resistor and (precision) OpAmp, that is the other way around.
Why do you need 130mA ?
The 130mA causes heat of 2Watt in the liquid Helium. That can't be good.
If only the resistance needs to be measured, then 1mA will also work.
The capacitor at Vref_out might leak, that might lower the reference voltage.
What kind of diode is that at Vout ? A Schottky diode or zener diode ? Most Schottky diodes leak and a zener diode leaks a lot, that will influence the voltage divider with 10k and 5k9. Could you set the gain to 0.5 and have the output in the range of 0...5V without voltage divider ?
The 10 ohm precision resistor is measured with a OpAmp and the GND is used as GND. I see trouble with the GND, since the 10 ohm is a low value. Better use a instrumentation amplifier as well and use a 4-point resistance measurement (in the same way as R2 is measured).
However, it will work. The Vref_out could be connected to an analog input, but also to AREF. When AREF is used, it automatically sets the reference according to the inaccurate current source. Will it be accurate ? Not very accurate, in my opinion. Every component adds more inaccuracy. Don't expect to measure with 1% accuracy.