You make your own decisions about batteries. All I can say is 850mA isn't much, and 400mA is less than half that. You make no mention of how long you expect this gear to run - or the power requirements of the peripherals.
All I am suggesting is that, if you use a bare 3.3v HC-05, you can use the same source to power it as you use to power a 3.3v Pro Mini. This may be either direct, or off Pro Mini's VCC pin. The bleeding obvious is that using a 3.6v breakout board with a 3.3v Pro Mini is a dumb idea, unless you have made proper provision to power it. This is simply a matter of using something that probably will work rather than hopefully might work, and you will not be the first to have a 3.6v board that fails when fed with 3.3v
Powering HC-05 off an Arduino signal pin can never be a good idea but using a signal pin to control a mosfet or w.h.y in order to control power supply to Bluetooth is common practice. HC-05 is not famous for its low power consumption, and I bet you haven't checked it, or how much power is available off pin A0. A Uno allows a max of 20mA and I don't suppose your Pro Mini is much different.
Jumpstart is an unknown term but I would have though there is no "seem" about HC-05 going dead when the battery dies, you should be able to bet your grandmother's pension on it. Further, there is nothing to suggest you won't have Bluetooth running in the proper manner as soon as you provide it with the proper power.
This may well be down more to luck than anything else, and your only problem is to introduce some certainty...