Giving 6VDC to the VIN on a 3.3V Pro Mini sounds like the way to go. The voltage regulator needs some room. You'd need to give about 7.5VDC to the VIN of a 5V Pro Mini, or the regulator would be spotty. Remember that batteries drop voltage as they are used, so design up from the "dead" voltage, not down from the "full" ideal voltage of the battery.