OK, here's my suggested circuit.
This shows a single stage. The 1.2V reference is included, and its output voltage can be fed to other stages like this one.
I have specified an OP-07 op-amp which has a much tighter input offset voltage specification than the MCP604x. This is important for accuracy at the relatively low voltages that the circuit deals with. You can use a trimpot to adjust the offset voltage; see the data sheet.
It needs to be powered from a split supply. I suggest +/- 5V. Decoupling capacitors should be used from pin 7 to 0V and from pin 4 to 0V on each op-amp. Filtering capacitors may be appropriate to improve noise rejection but there's little point bypassing the feedback resistor with a capacitor because the stage gain is pretty low.
The circuit uses four resistors with critical values.
Rs is the input shunt resistor that converts the input current into a voltage on the non-inverting input. I have shown a 10 ohm resistor on the schematic, but values up to around 40 ohms can be used, and higher values will make the circuit less affected by the op-amp's input offset voltage.
Rf (feedback), Rg (ground) and Rb (bias) connect at a summing node at the inverting input of the op-amp. The ratios between their values are interrelated, and must be correct to give the correct gain and DC offset. Rf can be calculated fairly easily once Rb and Rg have been chosen, but I haven't figured out how to calculate the ratio of Rb to Rg. It may be beyond my maths ability to give you a formula for that ratio.
Here are some explanations and calculations.
The difference in voltage at the op-amp's non-inverting ("+") input, over the 4~20 mA input current range, I will call dIN. It's defined as Rs * 0.016.
The gain of the amplifier I will call G. For an output range of 0~1V, G needs to be 1 / dIN.
G is determined by the ratio of Rf to (Rb || Rg). (The expression Rb || Rg means Rb in parallel with Rg, and can be calculated as 1 / ((1/Rb) + (1/Rg)).)
Specifically, you can calculate the correct value for Rf by calculating (Rb || Rg) and multiplying it by G-1.
As for calculating the ratio of Rb to Rg, I'm not sure how to do that. My maths skills are probably not up to working it out either. I used trial and error and the attached spreadsheet.
Here are a few Rb values for different values of Rs (the input shunt resistance). These all assume Rg is 1000 ohms:
Rs = 10 ohms; Rb = 24200 ohms.
Rs = 20 ohms; Rb = 9200 ohms.
Rs = 22 ohms; Rb = 7836.36363636 ohms.
Rs = 33 ohms; Rb = 3290.9090909091 ohms.
Rs = 47 ohms; Rb = 582.9787234043 ohms.
Rs = 50 ohms; Rb = 200 ohms.
In all of these cases, Rf needs to be changed as well; the attached spreadsheet calculates the value for Rf according to the formula above.