Hello and welcome to just answer. Yes the battery voltage needs to be at least 12.6 volts, and should remain close to that when a load is applied like turning on the lights. I am thinking that you may have an open cell in the battery itself, and no matter how much you charge it the battery will not work. What I recommend doing is to pull the battery and have it tested at your local auto parts store using an electronic battery tester. Most auto parts sores will do this for free. Replace the battery if it fails the test. Sometimes batteries will fail completely if a cell goes open, and I think this is what has happened to yours.
I hope this information helps, and let me know if you need further assistance with this.