To my knowledge both areas, and others, remained productive into the fifth century. That's why the barbarians were so eager to take North Africa. Egypt already was a desert except for the Nile which originated far to the south in subsaharan Africa, where cooling wasn't likely to be a problem.
Absolute prosperity=/=relative prosperity. North Africa became less productive but still remained productive overall.
It was about 10% around c 300 CE and skyrocketed before those incursions, in the decade or so after Julian. From what I've read, christianity was in a dominant position in the Empire in the latter fourth century. This appears to correlate with military weakness, which invited invasion. In other words christianity was more of a cause than an effect of the invasions (or inability to deal with them effectively).
So why was the East able to deal with them effectively? Oh, right, because they had the Egyptian breadbasket and economic powerhouse, which went relatively unaffected by climate change as you point out.
Lol, I don't think that "support" amounted to much if anything. It didn't mean more revenue for the state, if the church was tax exempt. It obviously didn't mean more citizen recruits for the army.....
The Church often wasn't tax-exempt, or at least was obligated to provide donations to the State on occasion. It also took over for the State bureaucracy in many cases, decreasing administrative costs.
But if economic crises can cost the State legitimacy why didn't this happen in the third century, with its rampant inflation and debasement of coinage? Add to that the economic effects of the plague, and mass plundering incursions...From a purely economic point of view, the Empire appeared better off in 400 CE than in say 270, or certainly no worse, but it just couldn't bounce back like it did previously.
Never said it didn't. Indeed, a lack of confidence in the Empire brought on in no small part by the economic crises was probably what motivated many soldiers to declare their generals Emperor.
And saying that 400 CE was a better year economically for the empire, whilst technically true, ignores the increased instability of the Empire's political and economic regime.