SIM800 or SIM900 library for Arduino
Most likely, the Serial3 buffer is being overrun because you are not reading fast enough. That is why when you use a simple loop, everything works and you get the whole message; very little time passes between reads. The simplest solution to this is to increase the size of the Serial buffer to, maybe, 256 bytes. Search for HardwareSerial.h in your Arduino installation and look for the #defined variable SERIAL_RX_BUFFER_SIZE in the file near the top. Change its value to 256 and save. This should provide enough buffer space to store the entire SMS until you read it.
Alternatively you could change your CNMI settings to push only SMS notifications, which are a lot smaller in size, to the Arduino. Then you can parse the notifications, extract the SMS index and then request the SMS from the SIM800 using the index whenever you wish, making sure to read quickly (you will need to craft a while loop to do this efficiently, with timeouts and all).