Regardless of the operational problems and diplomatic costs the Soviets would face, them moving and seizing the Kars-Ardahan 1914 border is surely as achievable as their historic gains against Finland, the Baltics, Poland and Romania.
The straits are much more operationally challenging because of the need to cross water and the more densely populated and urban terrain.
The Soviets operational solution to their strategic goal may be to maximize their advance as much as possible in northeastern Turkey, say to the line they occupied by OTL early 1917, and then offer to return territory up to the 1914 border line in exchange for the Turks accepting Soviet establishment of forts and naval bases at the straits.
{Soviet life gets more complicated if the Turks hold out, say "no deal" and otherwise do not behave as Moscow hopes and expects)
In theory, this type of "Finnish-style" victory (not sure it would be as hard fought for the Soviet side) would leave Turkey motivated to fight a continuation war, once the Germans begin Barbarossa. But the problem for the Turks at the point would be their southern frontiers and coasts would be vulnerable to retaliation from the WAllies in a way that Finland's Baltic frontiers were not.
As for the Romanians and Bulgarians - the Romanians will be freaked out but will have little ability to do anything to oppose the Soviets that would not simply bring greater harm upon themselves.
The Bulgarians will have mixed feelings about Russia in the straits, balancing their Russophilia against their monarchism, discomfort with another major power's control of their trade and their own desires for Constantinople and the straits.
---It probably would not go down this way, but the Greeks, Bulgarians and Italians could try to join the USSR in an anti-Turkish pile-on to all get their own slice.
--When Barbarossa rolls around they abandon what they have in Turkey and end up at significantly worse footing, if they didn't get trashed immediately.
This might be hard in practice- the Soviet presence in the straits, even if not a full occupation, may give the Bulgarians the stones to not align with the Axis and permit German forces to pass. This of course carries a high risk of war for Bulgaria, but it slows down everything the Axis can do in the Balkans a little bit.
Also, western hostility to the Soviet occupation of the straits evaporates entirely once Germany invades the Soviet Union and thereby becomes an ally. London immediately comes to regard the straits as a valuable flank for the anti-German coalition to keep a hold of.