Not possible to send an sms in the background. It might be possible to send the text in the Foreground.
I am aware of two possibilities:
This extension might be able to send an sms when your screen shuts down in the foreground. I built Foreground apps to use with the Pedometer and Texting using Uliss' previous version of the extension. That extension is now deprecated. I have not yet attempted to make a version using his new version of the extension.