No I don’t but…
As long as the power supply can support 2A peaks (!) with voltage above 3.20 volts, it shouldn’t matter if you use a linear or a switched regulator. But, in general, you’ll want to use:
main power --> power regulator cca 4.1V --> backup battery --> capacitors --> GSM module
with wires, MOS transistors or diodes for “–>” as necessary. If you want to replace the battery with capacitors, they have to be VERY large. Even with the battery, we are using 470uF.
You should test if adding the battery helps. Also, make sure you are using a good antenna, it may reduce both EM interference and power requirements. If it works, go back to small antenna.