A GSM module will need around 1,000 mA at times. Maybe more.
Personally I would choose a valve regulated lead-acid battery rather than a lithium one. VRLAs are well suited to continuous float charge and long standby use.
You can charge it with a cheap 12V charge controller or build something with a regulator. I have a solar powered Arduino project and I just use an LM317 adjusted to 13.5V as a charge controller.