Dear Nick,
I figured out how to make this circuit work MUCH better.
Previously, I had connected the Pro Mini RAW pin and the HC-06 Vcc pin both directly to the 3.7V Li-Ion battery. When the 3.7V Li-Ion battery was fully charged, it was somewhere around 4.4V if my memory serves me correctly.
The problem was that the HC-06 would power down at about 3.6V (Vcc for the HC-06 is 3.6V to 6.0V). So while the HC-06 would power down, the Pro Mini would still be running. This was not good.
I rewired a couple of things. First of all, I removed the power connection between the 3.7V Li-Ion battery and the Vcc pin on the HC-06.
Then I routed a wire from the Pro Mini Vcc pin to the Vout pin of the voltage regulator on the HC-06. Good reference at this link: https://www.instructables.com/How-to-Repair-HC-06-or-HC-05-Bluetooth-Module/ . Go about half way down and it shows you how to identify the 3.3V regulator and which pin is Vout.
NOW both devices are running off of the same regulated voltage source. They work reliably and the connectivity is excellent.
--Neal