Thermocouple signal conditioning - how to deal with low output swing margin of amplifier?
Note that the 20mv error is a measure of how close the output can go to 0v (using your supplies) against a 100 kilohm load pulling the output to "mid-supply" i.e. 2.5V.

Without that pull-up, as in your application, the output will go considerably closer to 0V.

However if you need to inject offset; the place to do it is "ref" i.e. pin 6 - whatever voltage is there, is the "0V" reference point for the output.

Note also that the thermocouple will ONLY generate 0V at ambient if it is actually a pair of thermocouples; connected back to back, one measuring the temperature of interest, the other being held at ambient (or 0C or some known point) to act as a "cold reference". (This may be what your "thermocouple" does, but its worth mentioning because the circuit doesn't show it). 

Otherwise you have at least three thermocouples in the circuit; two of them being unknown quantities, formed by the connector pins. In which case the actual voltage is anybody's guess.

One further point - I see you connect the thermocouple via a connector. It's worth considering what happens when unplugged : I recommend a high value resistor from -in to +in to prevent +in floating. (1 megohm with the 1na bias current implies an offset of 1 mv with no thermocouple, and it will have no measurable effect with the thermocouple present)