What would the Byzantine Empire have to do to survive?

[Theodosius the Great]

That would have been the best way to survive - get access again to the western soldiers!
The western legions were always of better quality than the Syrian troops. Time and again we hear of the supposed higher quality of the gallic infantryman when compared to the Syrian troops. The posssibility of the Roman empire to move legions around was one of its big strengths.

They still existed at that time, Procopius tells us. How could the empire link up with them then - via the Alps from reconquered Italy ?

I can't see any possibilty of how the Romans would have been able to 'crush Islam' after winning the battle of Yarmuk. They would have retained Syria (at that moment) but they still would have lost Egypt and the other African provinces. But the Arabs were also busy crushing Persia, which the Romans would hardly have been able (or willing) to stop. My guess is that even if the Romans would have been able to hold on to all eastern territories (which I doubt they could), the Arabs would have replaced the persians as enemies. Islam would never have been 'stopped' by the Romans.

But Egypt fell after Syria. The former was more difficult to conquer than the latter.
Depending on how great a victory the Byzantines could have achieved at Yurmuk, the Arabs may have been rendered too weak to conquer all of Persia, IMO.

Too much traditionalism still remained for the Romans to have tried to "crush" a religious system in another culture. While they held Christianity to be the state religion at that time, they would probably still have let others keep their own. But we're talking late 7th early 8th Cent AD here, aren't we?

Not if Islam was seen as a heresy which it probably was given a shared belief in Christ (to a point), the Old Testament, etc... But I don’t think Killaman meant a conscious, concerted effort to eliminate it. Rather it would have imploded on its own by losing credibility if the Arabs were massacred at Yurmak, for example. (Just think, if Constantine was defeated at the Milvian Bridge would Christianity have spread as fast as it did in the fourth century ?)

~Theo