3.3 powered microcontroller interface with 5V powered ADC
To answer your question: I would use the 3V operation with the 1.225V reference so you can use less power with your controller. Here's my reasoning:

A/D resolution is usually specified in bits.

Let's denote the largest input allowed as FSR (full-scale-range). The FSR is often, but not always, determined directly by the reference voltage so often FSR and reference voltage are used interchangeably. It's the FSR we care about. If we define the A/D resolution as the smallest input difference that can be detected then the resolution is FSR/2^bits. For a 16 bit converter resolution is FSR/2^16. A 12-bit converter has a larger (poorer) resolution of FSR/2^12.

(Incidentally, the smallest input difference that can be detected is also obviously the smallest input that can be detected).

Therefore absolute resolution can be improved by using a converter with more bits, or by using a smaller FSR. However in my experience, choosing a small FSR A/D is rarely done. Instead, if optimal resolution is required, one scales (amplifies or divides) the input signal before the A/D so that the largest expected input signal is just less than FSR.

For example, if your input sensor will output a maximum of 300 mVolt after the I/V conversion, you would amplify the signal by 8 for an A/D with a 2.5 Volt FSR. (I've left a little headroom to avoid clipping). You could also divide a 12 Volt input signal by 5 and use the same A/D. In summary, the FSR doesn't need to matter if you simply scale your input signal before the A/D.

Note that the A/D converter you refer to in your question has an internal, programmable gain amplifier so you may be able to handle the scaling without any external parts.