Now, if you want a Varangian on the Throne, there may well be a way (albeit short-lived). Suggesting this as a good sport. IOTL, Harold was imprisoned for a time for what many believe was embezzlement and somehow finagled his way out of the situation and went on to earn various court dignities. Let him avoid these charges (somehow) and not face the subsequent marring of his name.
Now, when he participates in the campaign of Thessalonika with Michael IV, he is a little bit more prestigious, perhaps having additional court dignities upon him. The victory there won't hurt either.
Now, Zoe being who she was, she was suspicious of everyone around her- so Harald will be walking a fine line. Now, some people actually did dislike Zoe and her treatment of people at court. Following Michael's death (as a victor), there may well be some opposition to her subsequent marriage to Constantine. It was these pressures that led her to put Michael's nephew in the line of succession.
Well, that tangled cobweb of Zoe's mind, seeing a friendship between Harold, Maniakes (a prominent and treacherous general) John the Eunuch (Michael's brother, and a man with pull at court) and the late Michael IV, decides to act. She imprisons Michael V and John the Eunuch, strips Harold of his honors, and attempts to depose Maniakes as catepan of Italy (he refuses, therefore starting his OTL rebellion a bit earlier)
Zoe's actions earn her no friends- Maniakes was quite popular, the late Michael IV and there Michael V had support among the soldiers of the Western themes and John the Eunuch had his hand in every pocket.
Harold escapes the city of Constantinople and sets out upon the Black Sea, where he uses the supposedly massive fortunes he'd earned while a Guardsmen to raise a host- there were supposedly quite a number of Viking pirates and mercenaries in the region, though their days were nearly done, and Hardrada had no problem IOTL at cobbling together unsavory elements into an army. Perhaps he also calls in a favor from Yaroslavl the Wise of Kiev, whom he had worked with about a decade prior, and who was intent on lessening reliance on Byzantium by associating himself with Scandinavians.
And so, you can have a Viking host under Harold Hardrada sailing for Constantinople- and doing so under the pretense of putting Michael V upon the throne. I doubt such a usurpation would last long, especially if Hardrada tries to make himself Emperor in name. But it has a chance at working, by utilizing the divisions in Byzantine politics and making it something more than a palace hall coup de force. Don't know his actual chances at taking Constantinople.
The biggest threats to such a state would come from Anatolia and the Balkans- The themes of Greece were mostly cowed and would, in the above scenario, be loyal to Michael V. Italy would be held down by Maniakes. Anatolia, however, would not be loyal and would likely be suspect- they were the ones to suffer from Black Sea Vikings. And the Balkans had only recently risen in rebellion- so any weakness would be acted upon.
Thoughts?