Yes LM2596 is my choice too.
Back to my stage 1 testing... By using one 12V 21W bulb as load managed to create a flow of 2A so to benchmark my batteries.
Instead of using data logging for having data of the actual discharge curve, decided my first test to be more primitive by using just my eyes and a timer.
The battery pack managed to work at 2A for 60 minutes and after that mark and in the next three minutes the battery voltage started drop down below 12V amazingly fast, so fast which I did not manage to record the actual CUT-OFF voltage.
I thought that I would getting at list 75 minutes by it and I was relaxed by moving around in my workshop, but I did not.
My first impression about this circuit it is very positive, even by be that tiny the load of 2A for 60 minutes it was not enough even to heat any electronic part over it = frozen.
The batteries it did become hot by reaching a temperature of about 40C.
After CUT-OFF my next thought was to recharge the batteries at the suggested max battery voltage of 16.8V but my power supply measured 6A and I though that is best to wait first so the batteries to cool-down before the next charging cycle.
The battery cells does not have printed specifications over them, but my opinion now is that they are at a minimum 2000mAh, which translates that their health are good and they can be trusted.