The short answer is yes, however, you don't do it on the phone line itself you do it via a route pattern.
Create a new partiton maybe called "PrivatePSTN" or something like that.
Create a CSS that can access the new partition for outbound calls (and any other partitons that need to be accessed from the line. eg other IP phones partition, etc.)
Create a route pattern in the new partition and in that route pattern set the Calling Number parameters as desired.
Change the CSS on the line you wish to restrict and restart the phone.
I think this should do it.