Current measurement with shunt
To address the question regarding the simulation behavior, According to the datasheet it is a rail-to-rail op. amp. only in the output:



You may change your gain and V_ref (which is like an offset) to meet the input specification. (2.65+1.15)/2 = 1.9 V at the op. amp. inputs for the equilibrium could be a starting point.

I'm concerned with your floating reference with the batteries. You must control your common voltage. Your circuit should follow the measured circuit reference (maybe by powering your 0-3.3V circuit from the 12V with a DC-DC converter or linear regulator) or you may provide an isolation layer.

Op.amp. with finite open loop gain (otherwise ideal for DC) and matched resistors:



Same op. amp. and matched resistors, common mode voltage:



Now with slightly unmatched resistors:



It will be worse when you consider that the real op. amp. will also have a common mode gain and the input range has limits.

Common mode voltage does matter hence you must control it for proper measurement. Every simulation that considers an infinite resistance from the common battery terminal and your circuit misleads your design, since even a 1k resistor is enough to guarantee the input voltage potential.