Edit: the following no longer works…
Second attempt …
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Create a new trunk with the same settings as your existing trunk. Do not populate the registration string. This trunk will be dedicated to private outbound.
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Override the trunk dial options and use this instead:
Ttb(custom-privacy-header^s^1)
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In extensions_custom.conf add lines like this:
[custom-privacy-header]
exten => s,1,Set(PJSIP_HEADER(add,Privacy)=id)
exten => s,n,Return
Use this trunk for your private outbound route then reload the dialplan. Might work, didn’t test