Replacing 4 x 18650 batteries with 12 x AA batteries
Would it ENELOOP provide the same working hour?


The Amazon AA batteries are rated at 2000 mAh typical (if you believe Amazon and that's a "big if"). The Lithium in the picture is 38.5 Wh (watt hours) so, if you divide that by voltage (14.8 volts) you get an ampere hour rating of 2600 mAh and that's considerably more than you get from the AAs.

There are a few other things that also might upset things like the effective series impedance (ESR) of the 12 series AA batteries - it might mean that your circuit cannot function if it takes quite high and short pulses of current.


  The OEM has a special circuit on it and it is not easy to remove it


This makes it less certain that any other battery other than the original will be compatible. It might be a conventional BMS circuit but it might be something that provides other functionality.