I agree with Soverihn and Practical Lobster.
Another interesting idea how to bring the Buddhist Middle East about could be to have it start in Arabia, too. There was a lot of trade between Southern Arabia and India (although much of it was intermediary) even before the 7th century and the rise of Islam. Now, "Indianisation" and / or the spread of Buddhism occurred in the preceding centuries in such unlikely and heterogeneous places as Southeastern Central Asia, the jungles of South-East Asia, the islands of OTL Malaysia and Indonesia etc. It stuck with steppe nomads and thalassocracies alike, with tribal and half-urbanised societies, with pastoralist and agriculturalist societies, with literate (China, Korea) and illiterate ones (Malays) alike, and it jumped high linguistic and cultural borders.
Why not have it extend onto Southern Arabia, where religious changes were occurring anyway? From there, interaction could take it to the desert-dwelling Arabs up North (either through peaceful contact and conversion, or by means of a nomadic conquest of the rich kingdoms in the South and then cultural assimilation, or by any other means), slowly adapting more and more to the cultural environment of the Middle East.
If Arabia later still explodes into the entire region, it might take its own version of Buddhsim wherever it goes. (The question is, though, whether it would have the same amount of appeal to wider parts of the population that Islam had... you`d probably have to create some serious alterations... It´s not excluded, though. There were enough revolts led by Buddhist monks in medieval history, especially in China.)