Before you order that buck/boost converter, look at the efficiency at low-ish current, and the ~0.6mA draw in shutdown.
The ProMini's onboard 5volt regulator has a shutdown (enable) pin.
But... it's not available to the user without removing the smd chip and cutting the track underneath.
Google "UVLO chip" (UnderVoltageLockOut).
I didn't see anything worth mentioning, but you might have better luck.
Maybe you could measure battery voltage on startup, and choose "run" or "deep sleep".
Leo..