One approach is to use two converters in parallel, with different fixed-gain amplifiers for each. You use software to combine the two sets of samples into a single floating-point stream.
For example, you could have two 24-bit converters, one with a 10V full scale range and the other with a 1V full scale range. When the first one indicates that the signal is less than 1V, use the output of the second one. The software uses the outputs of both converters where the ranges overlap to maintain an accurate relative scaling factor between them.
The trick is to make sure the second converter (and its amplifier) recovers quickly from saturation when the signal goes outside its range. If the saturation characteristics are well-known, the software can help compensate for them.
Well, let's start with your range and accuracy specification (and circle back to some other things later):
5 grams out of 200kg full scale is a factor range of 200 / 0.005 = 40000 or put differently, that's 1/40000 = 0.0025%. That's a pretty tall order for noiselessness, so you will need some form of ranging, which you apply externally. Just amplifying the result of your load-cells will also amplify the noise, which will still give you the need for a below 0.0025% noise.
I'm not saying that's impossible, but it certainly is very highly impractical. In design time and component cost.
That leads me to Least Significant Bit enthusiasm, that you seem to have. You can get "approximately 1LSB" accuracy, which as you say is about 3g, but only with good averaging and heavy calibration. ADCs are always very slightly mistaken, they are offset a tiny bit, they are very very slightly not linear and they may also have a tiny bit of gain.
If you have the datasheet it will mention numbers for "non-linearity" and "offset error" and all such, they all combine through fancypantsy maths to get you a number of confidence expressed in an amount of LSB. I would, without looking at the datasheets (because I'm tired after a long day), not expect much better than 5LSB. That means that the measurement will be off by up to 15g at least. You can potentially calibrate that out, if you have the tools. (EDIT: Or as Olin suggests just go for the 20bit+ sigma/delta ADC)
This then leads me to advise you to find out what the specifications of the load-cells are at smaller numbers. I can very easily imagine the cells being manufactured to a 50g accuracy in a commercial scale, or even worse (250g wouldn't even surprise me). Which also means the manufacturer stopped looking at non-linearities and correcting them in the manufacturing process once they fell within a +/-25g range.
Then, there's the sharing of the load.
If you put the weight on 4 load-cells and each can handle a maximum of 50kg, what happens if you put 200kg on, but not EXACTLY in the middle of the square made by the cells? More weight goes to the one closest to the centre of mass. Usually a 200kg scale will contain load-cells at the least capable of 100kg each, if it's a professional one.
With regards to the reference voltage:
Do not use a simple 3.3V regulator, they are 5% off, and if it's a DC/DC switching type they will also increase the amount of HF noise, they are the worst possible choice for a reference voltage
The Vref pin of your processor will also not be very accurate, since the processor is made for processing, not for referencing. The reference is just a fun extra they added in. In some cases they can still be the same 5% off that a 3.3V regulator is, especially over temperature.
If you want high accuracy, get a low-noise, stable, high precision reference, either from a circuit using a programmable zener type thing, or just by buying a chip for it from Linear, TI or Analog Devices.
Best Answer
The datasheet for the ADS8320 gives an example circuit. See Figure 22 of the datasheet.
For reference figure 22 from the ADS8320 datasheet:
The given circuit doesn't quite match what you are doing, but it and the text give you a couple of pointers.
The 1nF capacitor is needed by the ADC input. Section 8.2.1.2 explains the reasoning.
From a practical standpoint, if you don't have it the input voltage will scitter around as it is being sampled and result in incorrect values.
The 100 Ohm resistor is needed because op-amps in general don't like driving capacitors. Putting that 100 Ohms in there prevents the op-amp from oscillating.
The 100 Ohm resistor and the 1nF also form a low pass filter. The given values have a cutoff of about 1.5MHz. That's not really low enough considering that the ADS8320 sample rate only goes up to 100kHz. If you can guarantee that the signal will not contain any frequencies above half of your sampling rate then this won't be a problem.
For your task, you need a rail to rail op-amp, a voltage divider, and the parts mentioned above.
That would look like this:
simulate this circuit – Schematic created using CircuitLab
The voltage divider (R1 and R2) brings the 12V sensor voltage down to something less than 3.3V because it uses standard resistor values. You should be able to figure out what ratio the divider really uses.
The two diodes are schottky diodes. They clamp the input to the op-amp to the limits that the datasheet for the LMP7716 gives. That's less than 0.3V above or below the power rails.
C1 is the 1nF required by the ADS8320 datasheet to keep the input from scittering around during the sampling.
R3 is the series resistor needed so that the LMP7716 can drive the 1nF without oscillating.
This is non-inverting unity gain buffer amplifier. I do not know why you think you need an inverting amplifier. That would be an unusual thing for the situation you describe.
Edited to add:
I've just noticed your comments on the frequency range.
You can add a capacitor in parallel with R2 to form a simple low pass filter. This will reduce the noise being sampled, and help avoid aliasing.
10nF should be sufficient (calculating the corner frequency is a task for you.)
Once you have it sampled, you can apply a filter in the digital domain and get better results than an analog filter would give you.
You need to get acquainted with Shannon's theorem and the Nyquist rate. This will help you avoid problems when sampling analog signals.