It sounds like you may have a handle on your question already, but I would like to add a couple of points about NIMH vs Alkaline.
You don't say how many watts or how many amps or milliamps your megaphone will be demanding, but the NiMH have a much lower internal resistance and so they can provide a much higher current without dropping their voltage as much as an Alkaline will. At higher loads, a Nimh will provide more power than a Alkaline; at very low loads, an Alkaline will provide more power (a very very loose generalization, but for example, an Alkaline will last forever in a remote control while the NiMH will not last as long - of course there is also the self discharge of many of the NiMH).
Also, most NiMH have a NOMINAL volatage of 1.2, but fully charged they start out at closer to 1.4 (I have measured some NiMH fully charged at 1.5) volts, and as mentioned already, hold to the 1.2 volts for most of their discharge.
The Alkalines often start closer to 1.6 volts, but quickly lose voltage as they discharge so that their average voltage through their life is about 1.2 Volts! Of course this depends on what the cut off voltage of your Megaphone is.
Don't forget that most NiMH have a very high self discharge rate and can lose 10-20% of their capacity in the first day, and 1% of their capacity per day just sitting there and are essentially completely discharged in 3 months or less! There are newer NiMH that hold 85% of their capacity over a year.
Check out http://batteryuniversity.com/learn/article/Nickel_based_batteries for more info.
As far as your 4 options, option 2 sounds like the best one. Options 3 and 4 have been discussed by others as to why they offer such poor performance.
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.
Best Answer
As the battery begins source current, it heats up and the chemicals inside begin to polarize, causing the internal resistance to increase significantly reducing it from the 15A theoretical to the realistic DC output of a few Amps max.
Chemistries like Lithium Ion can pump out far more amps for longer, because the chemistry does not break down so easily as those in Alkaline and Nikel Metal Hydride battery chemistries.
Another issue is the voltage of the AA battery while under such a huge load will sag significantly. Total rated capacity also takes a massive hit for certain chemistries, meaning sustained loads that large will drain the battery 5+ times faster than expected.
Again, Lithium Ion chemistries are more resilient to voltage sag and capacity loss under load. Unfortunately though, Lithium Ion batteries tend to explode into fiery awesomeness so care must be taken - with great power [density] comes great responsibility.