I was indeed talking about a push from Turkey here plus an attack from army Group B from the steppes. You are correct on the difficulties of the push, I just pointed out that it would be closer and with better strike distance from the airforce although I doubt the Germans would bomb Baku as they want it's infrastructure for themselves. It is also a front that tanks aren't as useful due to the mountainous terrain. Anyhow a double attack on that front could be troublesome for the Soviets but either way with the Turks having 3 different open fronts and a lot of enemy troops on each of them that are better supplied and equipped I don't see it happening.I might just have misunderstood what you meant with southern push though, could be that you meant that the German high command ignores Stalingrad to go for a southern push for Baku, but even undersupplied soviet soldiers will be able to give stiff resistance to the Germans in the mountains, perhaps even enough for the Soviets to use Stalingrad as a base to cut off the axis forces.
Generally speaking, as vital as the Caucasus is towards the defeat of the Soviets, any cause of action is likely to be incredibly risky. Although Hitler is cocky, I'd bet he would also imagine a northern or central push to be the most logical choice.