This guide was very helpful for me and allowed me to save my Ring Pro. The only thing I would say it is missing is a better description of how to get the black connectors off the board (two of the orange circles). They are extremely fragile and must be dealt with carefully. The first one I removed shattered and cost me additional work during the reassembly. The important part I learn is to not apply any force to wires themselves (don’t lift from the cabled as described). I found the best way to get the connector off the board is with a very small knife slid under the connector. I followed underneath the wires with the blade to slide it under the connector. Once the blade is under the connector you can gently pry it up and it will be reusable. lifting from the wires caused my connector to fall apart.
This is the battery worked for me. basically you are looking for a 200mAh 3.7v cell.
Amazon search - uxcell Power Supply DC 3.7V 200mAh 552025 Li-ion Rechargeable Lithium Polymer Li-Po Battery