A very small amount of the USSR's wartime commerce travelled through the Dardanelles in OTL, largely due to the Italian threat to Mediterranean commerce. As such, the German closing of the Straights would not actually impugn Societ ability to prosecute the war. For this plan to actually have any merit, Germany would have to seize the entire north of the country and attack the Caucasian oil fields. There are several mountains in the way which make the defence of those fields from the southwest reasonably simple.Then you missed my point, since I meant to say that Germany could withstand such a pre-emptive assault and end up better overall by invading Turkey, or even invade Turkey simultaneously with the USSR and still be better off. Especially considering the USSR would need to reroute troops to the Caucasus in such scenario.
However, even then, it might not be unimaginable that Stalin would have bought a German invasion of Turkey anyhow, and far from becoming hostile to Germany would instead be tempted to come to Germany's aid ala Poland, and take Turkish Armenia, which was something the USSR also considered doing after the war (though Turkish membership of the NATO prevented it).
All of this, of course, assumes Bulgarian support for an invasion of Turkey was necessary, which totally isn't the case. Bulgaria's support for the Axis in a military sense was completely negligible, and the Bulgarians were in because of political benefits for the Axis, and certainly not because of military ones.
While Germany may very well succeed in closing the Straights, all they would accomplish is opening another front and gaining themselves a new enemy, since they were not capable of actually taking and subduing Turkey, a not-insignificant military power - more so than any country Germany conquered, as a matter of fact, except France. Poland likely had a better military, but Turkey's natural defences would give it an edge. That extra front would enable Soviet forces to concentrate against the Germans in Romania, then Hungary. With Germany now deprived of oil, controlling the Dardanelles wouldn't mean much.
Stalin could not have permitted a Great Power - and Germany certainly qualified - control of the Straights. That was a pinnacle of Russian foreign policy for centuries, and while Stalin dropped several Czarist foreign policy issues, he was not about to let Germany control the Dardanelles. The Soviets would have launched an immediate assault on the Germans if they had tried to pull that. Armenia was itself merely a means of prodding Turkey into granting joint control of the Straights to the USSR.
How are those German troops getting to Turkey without crossing Bulgaria? Why would Bulgaria grant the Axis transit rights without a share of the spoils? And why would Hitler allow them spoils without forcing them to make even a token military effort?