Understanding load cell and instrumentation amplifier
Presumably your sensitivity of 3mV/V is with a 500 gramme load. and given that there is a small offset of -0.089 mV and this rises thru zero mV to 2.895 mV, there is a change of 2.984 mV - that's quite close to 3mV and, as always with mechanical-electrical devices you need to make a calibration adjustment. In other words the sensitivity of the device is nominally 3 mV per volt and 2.984 mV represents a small (and acceptable) error of 0.53%.

An In-amp is also subject to errors both in gain and offset. 3.203 mV offset represents an input offset of 0.032mV. Realistically, the error from the in-amp is adding to the natural loadcell error of -0.089mV and making it +0.032mV i.e. the in-amp input offset error is 0.057mV.

The device you have chosen has a maximum input offset error of 0.3mV at ambient temperature conditions so this error is well within specification for the device. 

As for the output not being 300mV but 278mV this is usually due to the tolerance on the resistor that sets the gain. It should be also noted that the gain accuracy for this device (assuming a perfect gain-set resistor) is +/-0.75%.

Other factors that can contribute to problems are layout, power supply levels, inaccuracies in your meter/measurement device etc...