Perhaps I’m not understanding what you mean. As I understand the situation as Turkey is not being invaded but would join the Axis camp because the Allies attacked Romania, who is part of the Axis.Why would an invaded country not adopt; "the enemy of my enemy, is my friend"? Finland is the case example.
The Causasus, Balkans and Levant is already 100 roiled, including Turkey which was only "technically" neutral . Turkey does not matter, if Ploesti is torched and ruined.
Turkey may not have much for armed forces at the time but anything piled on while North Africa is in a fluid situation and the Soviets are getting smashed left, right, and center isn’t a sound strategy. Opening multiple fronts doesn’t tend to work out well.