Then I'd just attach it to the 3.3v rail of the MDBT42Q breakout board with a diode. That way if the 3.3v rail ever drops (because the voltage source powering it fails) then the battery will take up the slack, but it won't be powering it normally.
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Ok, great.
Honestly, I'd just use a big non-rechargeable lithium battery. If you use an AA format non-rechargeable lithium you're looking at ~2500mAh and often a 10 year life: https://uk.farnell.com/c/batteries-chargers/batteries-non-rechargeable?battery-voltage=3.6v
Then I'd just attach it to the 3.3v rail of the MDBT42Q breakout board with a diode. That way if the 3.3v rail ever drops (because the voltage source powering it fails) then the battery will take up the slack, but it won't be powering it normally.