Scaling analog sensor output (0.4V to 2.7V) to full dynamic of ADC (Vref 3.3V)
Your signal range is, as a percentage of the range of the ADC, 71.2%. In equivalent LSbs that is 2917 LSBs. That's equivalent to a hypothetical ADC that has 11.5 bits resolution.

So my first observation is why bother: -

Given that any ADC has a real range that can never be relied upon to be exactly 0 volts to Vref, I don't really see much of an advantage to what you propose. 

But, say you used an op-amp that modified your signal to the range you believe you want, what errors are incurred in doing so? Are you going to use 0.1% resistors (10 bit accuracy) or maybe 0.01% resistors (a bit over 13 bit accuracy). Are you willing to pay for that accuracy?

Each resistor will incur an error and those errors can be assumed to accumulate. Plus, there is the error from the op-amp and it's inability to get within 10 mV to 50 mV of the supply range means you can never get the full 12 bits you desire.

So, do you stick with an 11.5 bit system or, do you design an op-amp (with significant cost for the resistors) that might give you 11.8 bits?

Or, do you go with a 14 bit ADC and forget about this silly talk of a level and gain shifter?