Powering a system using batteries I made an arduino nano IR sensor based auto liquid dispenser. Currently it is being powered by a 12V 2 amps adapter. The 12v adapter is connected to a LM2596 DC to DC buck converter which converts the 12V to 5V and powers the arduino, IR sensor and a regular green LED. Also I am using a RO valve solenoid (rated upto 24V DC) which is being powered by the 12V adapter. So whenever the IR sensor is triggered, the DC motor (5V 200mA) and solenoid (12V 300-500mA) are turned on for 3 seconds.

Now, I intend to power this system via batteries. I am considering the following 3 options.

a.) Lithium batteries in series (4 of 3.7V batteries). But again here I believe things would be complex as I have to make a BMS and charger for this. Is anything available off the shelf so that I can directly integrate the battery and charger with my existing system ?

b.) Connect 8 AAA batteries in series to get 12V. But how long would these batteries last for my application ?

c.) Connect 2 9V batteries in series. Again, can you guys please tell me how long these may last in the system.

Arduino nano I believe consumes 20mA and the IR sensor probably around another 10- 20mA.
If it wasnt for the solenoid, I was considering connecting a mobile power bank to power the system.
I think you should go with 3 lithium batteries in series and for proper balance charging of batteries, you can use an off the shell 3s lithium charging circuit like this here.So you will get 4.2*3 = 12.6V on full charge and on full discharge you will get 3.7*3=11.1v out of the battery. 

So still I think your pump motor would work on 11v and ther are chances that your solanoid also work on 11v it will just draw more current then usual.