What if Turkey joined the Anti-ComIntern?

I don't think it would have really mattered much, perhaps they could have helped the Italians a bit in their conquests, or like you said provide a shortcut to Stalingrad. From my take also, the Turkish government relied heavily on shipments from the U.S and Great Britain, such as naval vessels and aircraft. The military wasn't significantly large, and the most that Turkey probably would have been able to do would be to use their air force (comprising of about 8,500 men, 350 aircraft) to help the axis powers. I honestly think that domestic problems would limit Turkey from doing anything, perhaps even a civil war looming. But for military actions, like I said before possibly helping the Italians and at Stalingrad, but perhaps also helping German forces in the North African campaign. Turkey could have supplied Germany with much needed oil supplies, which would have been a huge game changer. That (being trade) would probably be the biggest contribution of Turkey in my opinion.