Low power is definetly not a point, the circuit is powered from a power supply. Ok, I will try to get and use the INA131 . Do you think the rest of the design is ok? I'm thinking of using additional guard traces etc.
The INA131 is horribly expensive though at $6 to $7 in 1000 off quantities, plus another $1 for the LMC6482. For less than half the price you could get a 24bit ADC with a x128 PGA with input referred noise of 65nV rms or less - eg. AD7797. The 16-bit version, AD7796 is even cheaper.
Using a sigma delta ADC means you can dispense with an anti-alias filter, or at least get away with a simple low pass RC filter. As David Hess pointed out, by using the ADC reference to excite the bridge, most of the gain and noise errors go away. A 2-layer PCB should be perfectly OK and save a bit more money. The main error sources beyond the strain guage itself will be the gain drift with temperature and integral non-linearity of the ADC (INL), and thermal EMFs.
Only you know your requirements though - if you're trying to develop an instrument to use to replace the standard kg or need high speed measurements then a cheap sigma-delta won't cut the mustard.
If you want really cheap then look at the < 40 cents 24 bit Chinese HX711 and 80 cents Nuvoton Technology NAU7802 or the 18 bit MCP3142 (< 60 cents in small quantities from China, but 8x max PGA gain).
Alternatively, consider a microcontroller with a sigma-delta ADC and PGA which should still cost a lot less than the INA131. Eg. the $2.3 STM32F373 has a 16bit convertor with 32x PGA (but the linearity isn't good at +/-200 to +/- 400ppm depending on reference voltage and PGA gain (ie. equivalent to 11 to 12 bits). Noise performance won't match dedicated ADC chips, but as they are quite fast at 50KSPS for a single channel, you can improve that considerably by averaging. TI, Freescale (now NXP), Silicon Labs amongst others have similar devices.
The Cypress PSOC 5 microcontrollers have a 20 bit sigma delta ADC with similar noise performance to the '373 but rather better maximum linearity error of +/- 32ppm (equiv. to 15 bits, but that is without the PGA - linearity with the PGA is unspecified). They are a lot more expensive though at approx $10.
[EDIT] Corrected AD7989 to AD7797 (AD7789 doesn't have a PGA). Added ADC/PGA gain drift with temperature as one of the major error sources.
One other advantage of the dedicated ADCs, such as the AD7797 is they typically include switches to allow the internal offsets of the ADC and PGA to be calibrated out when required. Using the instrumentation amplifier you would have to provide an analog switches or relays to short the amplifier inputs to provide the same functionality.
Many low cost ADCs have input multiplexors which would also allow you to calibrate out the gain errors/drift of the ADC and PGA by switching the input to a known reference voltage using voltage divider resistors connected to the same bridge excitation voltage. The resistors would obviously have to have appropriate precision and stability specifications. Doing the same with an external amp would require additional switches (and control signals), but that would allow a cheaper amp to be used as offset and gain drift would no longer be important.