First off, this is a VERY plausible option.
Not sure about distributed capital, that sounds too rational, the traditions and trends of that time favoured a dichotomy between very widely distributed, localised power (feudalism) and focalisation around the monarch and court. But Alexandria would sure have made for a huuuge capital.
Also, an Egypt-based ERE, if still emerging in the 4th century, would not necessarily be monophysite. Apollinaris and Eutyches weren`t exactly from Egypt, and a lot of the reasons why a a Coptic Church came into being as monophysite was because of the politically relevant geographical distance of Alexandria and its Patriarchate to the new political centre of a Chalcedonian ERE. With the centre in Alexandria, it´s just as likely that Egypt sticks with the same official doctrine as OTL while heterodoxies take root somewhere else.
As for strategic focuses, apart securing the Levante, Cilicia, the Cyrenaica, Africa and their respective desert or mountain hinterlands, (as well as Cyprus, Crete etc.), some degree of priority might be placed on the Red Sea and on a trade with India that bypasses the Sassanids. An Egypt-based empire can feed and defend itself, but if it wants to thrive politically and excel economically, it must rule the waves, and not just of the inland seas (mediterranean and black), but also the wide Indian Ocean; make converts there, establish new outposts, spread the gospel and the fame of the Imperator Basileous Melki among the black Africans and the Southern Indians etc. Southern Arabia would be an important bone of contention with the Sassanids, which would still be the no. 1 rival.
The post-Merotic kingdoms would also be really important, or maybe they wouldn`t even disintegrate.
All of this is rather bad news for the Roman- and Greek-speaking Danube and Balkans, though.