You can manipulate these settings in several places in FreePBX; and if necessary, manually editing Asterisk dialplan itself via modifications to /etc/asterisk/extensions_custom.conf
But before getting under the hood like that, can you try checking in the FreePBX GUI some settings such as:
Connectivity → Trunks → your_PJSIP_trunk_name → pjsip Settings → Advanced → Send RPID/PAI and marking it “Send P-Asserted-Identity header”
…then let us know if that helps. You may need to combine with changing Outbound Caller ID settings, either on a per-Extension basis, or in the options under these areas: