I’m not sure of the current position on SIP out of dialogue messaging in FreePBX, but I remember seeing people having to write custom configuration. Also note that there is little consistency between providers on how to get messages in and out of the public network, see for example: New open source SMS Connector module
Also, if you intend to send messages to US public mobile network numbers, you will need to register under their 10DLC rules.
I’ll leave it to someone more familiar with the process to tell you how to handle incoming out of dialogue messages, under FreePBX, as I would have to go back to Asterisk first principles.