Car Will Not Start problems of the 2012 Infiniti G37

I brought in my 2012 for maintenance at an Infiniti service dealer. I received a "new" battery as I was told the battery needed replacement soon. Come to find out, the "new" battery was actually a faulty component and my car would not start. I brought it in for diagnostics and they said nothing was wrong. I told them that it must be the battery since I never had that issue before. The battery issue continued and I would have to jump my car off. The day I had to bring it back to Infiniti, I had to jump off the car and right after I did, the airbag light began flashing and codes indicated a faulty diagnosis sensor unit for the airbag. They then did their diagnostics again and determined that the battery itself was the problem in the car-starting issue, yet the Infiniti bag light continued to blink. The dealer said the cost would be $1600 to replace the sensor/control unit. The airbag has never been deployed and I never had issues until the battery replacement with a faulty battery and thus creating the need to jump start the vehicle. The light came on right after a jump start. I never saw the light before that.