High Peak Current Voltage Regulator
You don't say exactly which GSM module you're using, but I used an idea (similar to that described by KyranF in the comments to your question) a few years ago with a GSM module when I had to deal with a very similar situation.

Similar to you I had to allow the module to draw bursts of current in the 1.5A to 2A range for short infrequent periods.

My module also had a fairly wide supply voltage range because it has its own internal voltage regulators, so I could tolerate a larger voltage droop over the course of the current pulse than would normally be acceptable - providing the voltage started off near the top of the module's acceptable range at the beginning of the pulse.

I experimented a bit with component values, and in the end settled on 3x470uF electrolytics in parallel and a 2.2ohm series resistor from my 5V 500mA supply.







simulate this circuit &ndash; Schematic created using CircuitLab

Of course in your case you'll have to use a 4.5V supply since that's the maximum your module is rated for.