The best job for this, is an old nokia. 3210, 3310 etc.
You can get into a message in a few short button presses on those phones, via the numeric keypad. Example:
Menu key, 2, 1 - will lead to create message for arguments sake
So map out the keys you want, and solder wires onto those keys on the keypad.
Then have your circuit, trigger those buttons and it will send your message, provided its a preset message. A non preset message will be a hassle.
This may not be exact, but you should get the idea
What phone do you plan to use?