GSM Battery/Amperage Usage
The breakout board clearly has a voltage regulator on it which explains the different voltage requirements.

The module itself wants ~4V. This is designed for use with cellphone batteries which output 2.9V-4.2V depending on their state of charge. It should work fine with any reasonably sized Li battery (tiny ones under 500mAh may not work).

However, you'll need to measure its power consumption in the "on and able to recieve SMS" state. I suspect this will be in the 100mA range based on my experience with similar modules. You'll need to work out exactly which level of "standby" this is for that module.

Achieving 3 months will either require that it's off most of the time (e.g. power up every 3 hours to check for SMS) or a large pack of many Ah.

Edit: look at the BOB schematic. First page has a spx29302 linear regulator from the power jack to regulated 3.8V. This is a convenience to let you power it off anything. The higher the input voltage the more will be wasted in the regulator.