I come to this thread in my usual capacity as a pessimistic naysayer.
From everything I've seen, the character records are very closely tied to the server from which they originate. The only instance I've seen of them appearing on another server was when they're copied (back in the test server days). So, at best I think, you might be able to copy your character to another server, then copy it back when you're done in the construct. But it's a fiddly process, wouldn't flow as well as the current transition to and from a construct, and would cause no end of problems in SOE's chat/e-mail/user system.