Power Efficient GSM modules
Peak current applies to very short pulses. Don't worry about this. In standby mode most GSM modules use much lower current (&lt;50mA).

Long time ago (8 years?) I was using Siemens XT65 GSM/GPRS module and it was pretty energy-efficient (energy consumption was similar/lower than typical GSM phone). There should be no problem to reach 5 day standby on 1000mAh 3.6V battery.

XT65 is very old model now and they probably not sell them anymore, but you can buy modern Siemens modules. They are not so easy to use and relatively expensive but I'm sure that energy efficiency is way better than in low cost modules.

Here is datasheet for XT65 module: link

It says that it uses up to 2A (peak), but this applies to bad signal conditions (when BTS is far).

If you want low cost device - buy some ancient Nokia phone (5110/6110). You can "talk" with them by using UART and AT commands. They can work about 1 week on 3Wh battery (800mAh/3.6V).