Primary Battery and DC Power Switch
After quite a bit of testing, it looks like the circuit design above is fine. The issue was with the Schottky Diode we had picked (Vishay VSKY05201006). It has some incredible characteristics (very low forward voltage drop, and tiny leakage current), but unfortunately, we had quite a few quality issues with it. The part is extremely easy to physically damage (found lots of cracks on the part), and due to its poor thermal characteristics, easy to damage during normal SMD reflow.