The complicating factor is that there are many versions of the "Surface Diagnostic Toolkit". You need to use the version intended for Commercial Repairs - which is available as a ZIP over here: [link|https://www.microsoft.com/en-us/download/details.aspx?id=100440|https://www.microsoft.com/en-us/download/details.aspx?id=100440|new_window=true]
When you run it, also make sure to choose 'Repair' instead of 'Diagnostic'. Then you will have the option to perform a Battery Verification. It will try to contact Microsoft servers to authenticate the new battery.