Hello BP,
Do you have any short term heavy loads on the battery, perhaps only for milliseconds? Being quite a small cell, its internal resistance may be significant, especially as the battery approaches its end of life. If your battery load is "peaky" you might need to consider a largish electrolytic capacitor (or larger than you currently have) at the supply connections to the MCU.
The series diode could also be problematic. This would need to be a Schottky type for lower voltage drop. What is the purpose of the diode - to provide reverse polarity protection, or to allow for connection of an external supply voltage? For a Schottky diode, you would need to allow a voltage drop in the region of 0.4 volts, and this may severely limit the battery life you will achieve.
In your case, another issue may be if the equipment is subject to very low temperatures - the battery voltage will further reduce.
Regards,
Mac