I soldered a prototype board implementing the transimpedance circuit I, Andrew and Rana discussed earlier.

PFA the schematic and picture of the board.

Some numbers:

Voltage rail: +- 15V
Transimpedance: Rated:16.549 kOhms (15k+1.05k+499)
                             Measured: 16.99 kOhms
Vol Ref; LT 1021DCS8-5 (Actual circuit would have ADR4550ARZ)
Measured voltage at Vol Ref Output: 5.003 V
Measured output at Room Temperature: 8.48 V
Temperature from the measured output: 299.005 K (25.9 )
                                                           It was certainly colder than that. I had no other good temperature reference.
Highest output on waving a hot air gun above circuit: 9.15V
Error in temperature perception from that: 358.5 mK

Possible sources of error:

• No bypassing capacitors were used. So some voltage noise from power supply must have crept into the circuit.
• Transimpedance resistor is made up of thin film resistors of 25ppm/K TCR. This translates to approximately 7.5 mK/K (change in perceived temperature per kelvin change in circuit temperature)
• The resistors in the adder circuit are thick film resistors however their effect should be minimal with even 200ppm/K translating to 1 mK/K.
• Voltage reference used was LT1021DCS8-5 with 3ppm/K TCR. This translated to 0.8 mK/K, so we should be good with this.

Methods:

For measuring room temperature, I first dabbed the AD590 with little-wet tissue to cool it down. This was done to make sure no remnant heat from previous runs is remaining. Then I put the AD590 inside a shallow metal drawer opposite the soldering bench and waited for some time to let the reading settle. I assumed the metal drawer must be in equilibrium with room temperature and must have high heat capacity to work as a good room temperature bath for AD590.

When seeing the changes due to circuit temperature change, I waved a hot air gun over the circuit so that hot air blows over the circuit and not onto it. The temperature of the hot air gun was 110  with the highest speed setting. I have no exact knowledge of how much circuit temperature I increased but it got pretty hot to touch by the time I reached a ceiling in how much output value is changing (about 9.15 V)

Conclusions:

I think I should go ahead with designing a layout of the circuit. There is only so much we can do in the design of the circuit itself. I have chosen 0.2ppm/K resistors for transimpedance to reduce our major source of drift. For voltage reference, since we are getting more precise ones in the same cost, I chose ADR4550ARZ (BRZ version on Digikey was out of stock) which has lower output voltage noise than LT1021. I have also ordered thin film resistors for the adder stage to completely eliminate the effect of drift from them.