This was something I thought could be done when I first got into VoIP, but I have learned that it doesn't work like that.
What seems to happen is that as soon as your call reaches the PSTN, the caller-name field is dropped, because most of the network can't handle it. So instead, the receiving office uses CNAM to turn the caller-id number into a name, which is displayed. The only control you have is that you can usually have your DID provider update the CNAM databases for you. This takes a few days, so you can't change your name dynamically.
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Steve