If its a standalone PBX I cant think of a clever way to pull that off. I think I know what you want. If you want your phone to act like its at another site you could use a remote cover path to have any calls that go tour current site show up at another location, but that wont make your outbound calls look like tey are coming from that other phone number. I've only seen that done with phones hanging off gateways, or with software like IP Agent.
We use IP agent software to make our Helpdesk agents in the Netherlands appear as if they are here in the US. Calls to that group in Europe come to a US number and the caller doesnt know the difference. When they make calls it appears to come from the US. But you need a license, and software to do this. You'd need to be able to authenticate to the remote PBX over IP. If this sounds like something close to what you are after you might want to reach out to your BP and discuss this further.
I cant think of anything else....Can anyone else think of something?