I don't really know if this is possible, but the topic is pretty self explanatory...
It is very frustrating when the PlayStation Network is down, and I can't access my own media. It would be great if the Plex app would work regardless of the PSN status which is always a bit flaky at best.
I'm assuming the XBOne and XBox Live work the same way...
These apps are Preview Pass apps. We've already heard from the Dev Team, that as soon as these apps get published for public consumption that the associated logins for each would be changing.
Right now, the account that is tied to the client app needs validating against some authority. That authority is the manufacturer's live network. The mechanism for doing this WITHIN the client apps aren't there to do this validating to plex.tv YET.