Hello,
Replacing a battery on a Surface is quite difficult and can be risky because of the internal design of the surface. If you just want a battery replacement without all the hassle and risk I'd recommend sending it to Microsoft to have the service done by professionals. Information on that can be found here:
https://support.microsoft.com/en-us/help/402352...
If you're set on servicing it yourself, as some are, there is a published guide on replacing the battery in a Surface (Note: It is a third-party guide, Microsoft does not recommend attempting the fix yourself and attempting the repair comes with no protection or warranty). This guide (https://www.ifixit.com/Guide/Microsoft+Surface+...) provides step-by-step instruction on replacing the battery, and allows you to purchase the necessary tools for doing so. Some online vendors sell batteries for the Surface, so a quick search should find a few retailers for you to choose from.
Overall, replacing the battery in a Surface 3 is doable at home, and from what I found can be done with a Pro 3 battery, however it is very difficult and you risk breaking the entire device from attempting it. My advice would be to seek professional service, because the cost savings of doing it yourself don't outweigh the risk of losing the entire device in my opinion.
I hope this has helped, and best of luck on getting your Surface back in good working order!