Capacitor for reducing voltage drops
I designed in the past quite a few boards with a GSM module on board. The boards were powered at 13.8 Volt.
I mainly used: Siemens TC35 and Gemalto BGS2 and others.
A FAE engineer at Siemens, back in 1999 told me: &quot;Power supply is crucial for RF modems. If the voltage supply is not stable, the module refuses to transmit in order not to transmit unwanted harmonics&quot;.
In the first generation of boards I had one LM2596 DC/DC converter at 5 Volt for both the module and the micro controller. Back then, the maximum voltage for the phone was 6 Volt.
In the second generation of boards I had two LM317 DCYR linear regulators able to deliver 1.5 A. The first regulator was at 4.2 Volt for the GSM module and the second at 3.3 Volt for the micro controller. I had a 6-layer board and used one internale plane to dissipate heat. After all, my boards made 1 phone call at a month. The maximum temperature I saw during phone calls was 65 Celsius degrees. Linear regulator are GREAT.
I've never had problems since 1999.
GSM modems are certified by manufacturers at the power supply voltage of 3.8 V.
If the power supply voltage is not stable during TDMA bursts the module automatically turns-off due to voltage sags.

To answer your question:
I assume here that your board is done and you can't add a second regulator.
Raise up the power supply voltage of the GSM module to 3.8 V or 4.2 V.
Use 1 or 2 Schottky diodes to generate 3 V for your microcontroller.
See what happens.