LM4040 AREF Pin Voltage Drop

No need to have an external reference for that.
The Nano has a stable internal voltage reference of about 1.1volt, which you can select with code.
All you do is divide battery voltage down to about 1volt with a two-resistor voltage divider,
and add this line to setup().
analogReference(INTERNAL);
More info here.
Tell us which battery if you need help with resistor values.
Leo..