Let's backup here a bit. The issue is the system shutting down when the power is low without notification. This is where the SMC services stores the battery status. Did you try resetting it as well as cycle your battery a few times so the value within SMC was correct?
First download this great little app: [http://www.coconut-flavour.com/coconutbattery/|coconutBattery]. Here's a bit more: [http://www.macworld.co.uk/how-to/mac/how-fix-macbook-battery-problems-replace-mac-battery-3522985/|How to diagnose MacBook battery problems, and replace a MacBook battery]
Then follow this Apple T/N: [https://support.apple.com/en-us/HT201295|Resetting the System Management Controller (SMC) on your Mac].