I would combine this idea with the "faction change DLC" idea I sometimes dust off when talking about changing alliance.

To recap - I think the best way to change factions would be an "spycraft" themed DLC that only unlocks after completing all of cadwells (or at least after completing the main storyline and beating Molag Bal), with a "loyalty called into question" theme, and a big choice in the middle of the storyline that determines how the story proceeds for your chatacter - stay loyal and prove your innocence, find the real traitor, etc.; or defect to another alliance, evading the agents sent agter you and grabbing some secrets to take with you on the way out.

Now, that premise is neat enough... however, there is nothing that would -require- your character to defect -towards- another alliance! It could also include a choice to, for example, defect to some other neutral faction!
(which would also neatly solve the "mainstroy" issue - if you defect to neutral after the mainstory, all is good again, yes? Then you start where you nstart, play until you kick Molag in his Bal, and then turn your back on the squabbling factions and go off on your own... Since the powers that be -definitely- would not spend the coin to program a fourth "mainstory" for "neutral faction characters"...)

Now, a neutral faction would have no game effects except for PvP in Cyrodil... and in there, characters of that faction would HAVE to be "yellow", as in neutral but attackable by -everyone-... (just like any strangers wandering a battlefield in medieval-ish times - no PvP-out-out hear!)
That of course may make their lives somewhat difficult, since everyone can gank them as they please, and we all know how some people might react to that (Yay, five AP! Kill!). What they would need was an safe base of their own... and that would be somewhat iffy, since the place is pretty much already equally divided between the various factions. Even so, there could be an hidden "neutral" location somewhere off to the side, or even several... a place for neutral characters to enter Cyrodil without getting ganked right away. Or several, there could be different "neutral" factions, even... a fighters guild outpost, a mages guild tower, a thieves guild cave, a pirate ship, a bandit camp, a traders outpost, whatever... such locations however would have feble defenses, and thus be prone to being raided by alliance warbands... which would fit well with the troubles of being neutral in a warzone, right?

The "mercanary" idea is a good one in that regard to add more options - neutral characters could temporarily join a faction, for example by equipping a tabard... which could happen automatically, you visit the mercenary contractor in the alliance capital, click the "yes, deal!" response, and get your "ally" status, that way mercenaries could not just switch sides in cyrodil... Such "ally tabards" then would then allow the mercenaries to be treeated as "friendly" by that faction, use their strongholds and IC bases, and neatly sidestep the problem... but could also be -unequipped- to play as "neutral" in cyrodil, and suffer the consequences when your former allies suddenly see you light up as attackable...