Apart from the lack of lumber you mention, there are a few other obstacles in the way of turning Aksum into a real Empire:
a) Aksum wasn`t much of a sea power. To keep its African and Arabian possessions united and to expand along the East African sea board, Aksum would have to turn into one. That requires lumber again, of course, but also lots of knowledge that would have to come from elsewhere. Perhaps a wave of persecution of Coptic Christians in Byzantine Egypt brings educated specialists into Aksum? A big advantage would be if there`d even be some military alchemists among them who know something about "Greek fire", so the Aksumite fleet could fend off pirates easily and control the Red Sea effectively?
Well, the Romans and Aksum were allies already - so gaining that knowledge isn't impossible. But there is certainly room that an investment towards controlling trade could be used to import lumber, to expand the fleet, etc - sure they wouldn't have a vast supply - but importing enough to make money and protect the fleet - invest that money in growing local lumber - and then Aksum can really start its ascension.
b) Aksum looked to Egypt for its state ideology instead of controlling it itself. Coptic Christianity had its leadership in Byzantine Egypt. A more autocephalous and perhaps even theologically distinct Aksumite Church should be actively proselytising. If you want to bring Arabia and East Africa under your control, you must unite it culturally. With episcopal sees and monasteries etc. used by the Aksumite church leadership as power factors and footholds in foreign places.
I'm going to assume you mean religion rather than ideology
I'm more interested by the idea of an Alexandrian flight from a dogmatic Constantinople, bu setting up a Patriarchate earlier could be possible - but would always be contested by Rome. If the King/Emperor of Aksum keeps the administration of a church tightly controlled - so that there are simply hordes of metropolitans under the Bishop of Aksum - then the sheer size of the Aksumite church would probably have to draw the Alexandrian Patriarchate to relocate to Aksum. A big moment - but if there is a seperation in ideology, and a particularly unapologetic Emperor - then a move to Aksum would be safest.
Otherwise yeah - it could also involve an Anti-Patriarch that could have been installed in some sort of treaty. (Perhaps the price of aid in a war with Persia?)
c) To make b) work, you must draw on a) and massively accumulate wealth from controlling the trade with India. A combination of wealth and faith might bring the necessary power and coherence.
There`s a reason why Abraha`s secession occurred and couldn`t be effectively prevented: the Aksumite centre didn`t have a far-enough reach and wasn`t enough of a magnet, metaphorically speaking.
So - we need wood, for trade ships, for more wood, for more trade ships, and then hopefully a home-grown timber industry - probably more likely to emerge as an accident from irrigating otherwise dry areas using earth or stone dams. Increased agricultural yield would help with increasing the population, and the trees likely to also be grown could be used for lumber.