Continuous battery monitoring during charging as well as discharging

An instrumentation amplifier would be able to offset the differential signal measured across the shunt. An AD8221 could run from smaller supplies (+/-2.5 volts) so this saves power immediately and its current consumption is about 1mA.

Your input signal is going to be in the tens of millivolts so no problem with the AD8221's limited common mode input range (on a split supply) and the AD8221 output swing is to within 1.2 volts of either supply rail so, from a +/-5 volt rail you can get an output that is +/-3.8 volts but you need an "offset" to bias it to a standard ADC input range of (say) 0V to 2.5 volts.

The reference pin on the AD8221 can be fed from a 1.25 volt reference to centre the offset from the AD8221 at 1.25 volts and make it suitable for a whole load of low power single supply ADCs. The ADC input would need to be protected from the AD8221 producing a negative voltage in case the signal from the shunt was bigger than expected but this is acheived by a simple series resistor.

Typically the AD7988-5 (a 16 bit ADC that I've used recently) consumes less than 1mW at a sample speed of around 100kSps so this would fit the bill but TI have very similar offerings as do LT. It has a maximum input current of over 100mA so a simple series resistor of about 1kohm is not going cause a problem BUT check the data sheets.

...as far as i have read about them (also their data sheets), if an ADC has both single and differential ended inputs, we can use it as a single ended input ADC OR we can use it as a differential ended input ADC. Which means we cannot use it as both single and differential input at the same time.

This is not always true. For example, I have used ADS1015 on a couple of projects recently. On this chip, whenever you switch the channel being read you also have the option to switch between single-ended and differential measurement. (This is not an endorsement of this chip for your project. Just an example of a chip that doesn't have the limitation you thought was universal)

Furthermore, even if you had a device that had to be configured as single-ended or differential for all channels at the same time, nothing prevents you from using ground as one of the inputs to a differential channel. So you could just configure it as differential and move on with your design. The only thing you'd lose is the opportunity to use the 4th input pin for some other purpose.

Another option, if you plan to use external signal conditioning, you can do differential to single-ended conversion in the signal conditioning circuit, and your ADC will never know the signals are anything but single-ended. This essentially makes the amplifiers shown in your diagrams to be external devices rather than internal to the ADC chip (and add some filtering in their feedback networks to reduce noise).