Would an early move of the capital from Rome to "Constantinople" or even Antioch in Hadrian's time strengthen the Roman cause i the East?
Every emperor, who tries to conquer and hold Mesopotamia, Armenia and perhaps Media, would spend the most time of his life in the East. Either in Antiochia or Nicomedia. If you look from Nikomedia for the best place to establish a fortified new capital, you find Byzantion.
An early division of the Roman Empire into a Western and an Eastern part?
Or divide the empire into a number of prefectures with the Parthian provinces as a prefecture of their own?
As discussed in the other threads about this topic, you need multiple strong leaders to manage such a huge empire. But this leads to usurpation in the roman world. Regardless if these leaders are caesars or stronger legates/prefects. The late empire had not much less usurpations than the principate. Just new types of usurpations evolved.
Well, if Odaenathus survives, he might conquer the East. There are vague hints, that he conquered Ctesiphon twice. But how long would he stay loyal to Rome? The option to become the new King of Kings yourself is just too tempting for every roman commander in this new big East.
I agree, that in the 3rd century this is even more a mission impossible than in the 1st or 2nd century. Diocletian stopped after the conquest of northern Mesopotamia by very good reasons.