I agree with this solution. It will work. Just set the ADS1115 to the lowest gain level / FSR. A bit of amplification would optimize the resolution of the system, but without this you'd already get something like 12-13 bits without much hassle using this setup. It'll likely be good enough anyway.
If/when using the ADS1115, I recommend Rob Tillaart's Arduino library; it's excellent. GitHub - RobTillaart/ADS1X15: Arduino library for ADS1015 = I2C 12 bit ADC and ADS1115 = I2C 16 bit ADC</t
If no ADS1115 or similar ADC is used that offers differential measurement out of the box, you can always try connecting the differential - lead to GND and the differential + to the ADC input. Since a thermopile pyranometer will only swing a positive signal anyway, it'll be safe and it may work fairly well, although attention to a proper grounding layout is necessary (separate analog signal ground from 'dirty' digital & power grounds, only connect them in 1 place)...but this is essential anyway when working with small analog signals.
I can imagine that looks fine. The question is if you'll actually achieve this. In a perfect world, you can actually make use of the 10/12 bits of ADC resolution you've got, but the world isn't perfect. This requires a very clean signal that's fed into the ADC and preferably at a lower impedance than the sensor itself will provide. So the amplification board you spoke of will prove to be useful, I think. What type did you get? Got a link to it or a datasheet?