Current sensor: averaging and feeding +/- signal into ADC
The best way IMO to accomplish that is to use an instrumentation op-amp with a ref input like the AD8220 which is designed for this purpose. They can take in differential voltage and output and offset-ed signal that can be directly connected to an ADC.
The gain is simply set by an external resistor and the output can be offset by setting a voltage on the ref pin. So you can easily have a differential input and output directly to a non-differential, grounded ADC. Those devices usually show very good linearity and performance. They are quite pricy but if you are not trying to shave every cent it would simplify your design a lot.
If needed, you can also use a simple resistor based voltage divider before the instrumentation opamp to accommodate the input range.
Be careful to the common-mode noise, better to connect one end of the sensor line to GND.
Note some design considerations, depending on the ADC spec, acquisition speed, and frequency of the signal, you might need a buffer op-amp between the instrumentation op-amp and the ADC to drive it properly in terms of current.
Concerning filtering, the simplest is simply to put an R-C filter after the instrumentation opamp and before the ADC, if you have a buffer opamp, you can have an R-C filter after each opamp.
You can also set an R-C filter before the instrumentation op-amp.
Last but not least, make sure your current sensor works in DC mode, hall effect current sensor usually do, but it is not specifically mentioned on the datasheet. Hall sensors for current sense are not very accurate, they have very strong temperature dependencies. They are usually needed for high voltage applications when you don't want an intrusive connection. If you deal with a battery, why not using a regular (4 pin kelvin) shunt resistor? it is cheaper and much more precise.
I would draw a schematic but the instrument op-amp doesn't exist in the StackExchange electronic drawing library.