The ADC is a 12 bit one and I will be happy if I can get to use all the 12 bits effectively. .... Not to be too greedy, I will be happy if I can freeze the reading to the nearest 0.1 Bar which i feel is somewhere between 11 and 12 bits.
Assuming your signal conditioner (SC) amplify the signal to 5V full scale, one bit will be 1.25mV. To even have a chance with the setup as you describe it, the SC must have a VERY low output impedance. I still suggest that you move the A/D to the SC and use a <$1 uC (LPC) to control that box. Do remember that if you are allowed to update less frequently averaging the readings can increase your accuracy dramatically.

Erik