The INA333 is relatively slow and also quite noisy, though as an AZ type essentially without 1/f noise. This can be a problem for a fast application as the noise BW is relatively high for a strain gauge use. If only a quarter bridge is used the drift will be large and thus no AZ type really needed and DC performance limited anyway.
With a typical maximum of 1 mV/V and maybe 5 V at the bridge, the full scale range of the sensor would be some 5 mV. Resolving 10 Bits would mean measuring to some 5 µV. 5 V at the bridge would already be high for only 120 Ohms.
For 10 kSPS samplig the BW would likely be at some 3 kHz (up to 5 kHz). So the needed noise level for 10 Bit performance would be at around
5 µV / Sqrt(3 kHz) = 90 nV/sqrt(Hz). So at best the INA with a 50 nV/sqrt(Hz) noise level would give an 11 bit noise-level.