Hi Chris, nice to follow your progress.
I am not sure the MCP73831 Li-Polymer Charge Management Controller can be used in that configuration.
The MCP73831 controls Current and Voltage to protect the battery , if there are other loads in parallel (the
the charger may behave strangely and the battery will suffer.
In this application the design prevents this situation, the battery is not charging and used at the same time, for that, there are other IC's:
http://ww1.microchip.com/downloads/en/appnotes/01149c.pdfTry to use a fresh battery and keep the power switch off while charging, the battery should behave properly then.
regards,
Vicente