First, the idea of attacking through Iran to hit the Caucuses and India is pretty ridiculous. It's all very mountainous terrain, and the infrastructure is terrible, not to mention hundreds of miles from a port that can supply them. Remember that even if you have the Suez, Britain still has Aden and the Horn. No convoy would get through the Red Sea. Going from Beirut to Baku would be more distance than Konigsberg to Moscow. Beirut to Karachi is even more of a ridiculous target. Second, again, Iran was not neutral in 1942, the very earliest you could even pretend that the Axis would be able to reach them. It was occupied, and if the Germans or Italians got to the borders, they would be just facing the Red Army in some of the worst terrain you could attack through.
I usually hold the shortest defensive line between the Caspian Sea and Indian Ocean and leave it there until it's mop up time for the UK, in which case the entire ME and the Horn is under axis control. We could make another thread on how India isn't worth it due to how stupidly undervalued it is. It also doesn't matter if Iran is neutral or not, if you are Axis you can declare war on them as you wish, I don't see it's neutrality as any relevance.
I send mountain troops to take it since that is the area they will do the most fighting until Army Group South Arrives, with some AT guns and Artillery, in case the Soviets want to do something really stupid and suicidal. Put a logistics wizard there and take the port there on the Caucuses (can't remember it's name) if you have to, the supply issue isn't that bad, unless you intend to send Panzers and such screaming through in large numbers.
I also did not say that my entire attack was focused around one main thrust through the Caucasus, that would be idiotic, the point I made was the take the oil fields. From there you can wait in a defensive line in the mountain until Army Group Center arrives.