Electronic – AC/DC 4W vs 2A GSM transmission bursts

The GSM shield looks to pass the input directly to a LDO, and regulate it down to 3.8V? I’ve no idea why they claim such a high input voltage requirement, I could be missing something.

With care you could patch the VIN & VBAT leads together on the module, and power them both from 3.8V. The LDO should drop itself out of the circuit, although check it’s not getting hot (e.g. dissipating any power). That device doesn’t explicitly state it can be back-powered like some LDOs will, but if you tie VIN & VOUT together it should prevent damage.

No promises though, if you break anything taking my advice I’m not liable!

But this would leave you with all logic powered @ a lower voltage.

One problem I can see in your code is the following line:

while (mySerial.available() < 0) {

The available() method returns the number of characters available and will never be negative, so that means your AT command will never be transmitted, in fact your main code probably isn't starting up. Also you're not terminating the command with a carriage return so try the following code:

while (mySerial.available() == 0) {
  mySerial.println("AT");
  delay(2000);
}

Often when playing around with a new GSM module I find it more convenient to be able to type things from a PC to check operation, so maybe for a start you could use something like this to pass data through:

void loop() {
  if (mySerial.available() > 0)
    Serial.write(mySerial.read());
  if (Serial.available() > 0)
    mySerial.write(Serial.read());
}

I used a very similar board recently with a SIM900 loaded that didn't come with much in the way of documentation and while searching I found some Arduino boards are set for auto-baud and some are fixed at 19200, so using a baud rate of 19200 that should work either way might be your best starting point.