It looks like you need to achieve:
Privacy: Id
P-Asserted-Identity: “don’t care” acceptable-callerid@acceptable-domain-name
I don’t know how you do that in FreePBX other than custom dialplan (to set the Asterisk option for privacy, not to directly add the privacy header). However there may be a more GUI friendly way.
I don’t think “hidden” has any special meaning.
“don’t care” means that this value is not critical rather than should be entered literally.
From might need to be the same, but their documentation does say that PAI has highest priority for caller ID.