I’m pretty sure that CVN-65 won’t meet the displacement bar.

https://en.wikipedia.org/wiki/USS_Enterprise_(CVN-65)

Displacement: 93,284-long-ton (94,781 t) full load[3]

https://en.wikipedia.org/wiki/Montreux_Convention_Regarding_the_Regime_of_the_Straits

https://en.wikisource.org/wiki/Montreux_Convention

The maximum aggregate tonnage of all foreign naval forces which may be in course of transit through the Straits shall not exceed 15,000 tons, except in the cases provided for in Article 11 and in Annex III to the present Convention.

Article 11.

Black Sea Powers may send through the Straits capital ships of a tonnage greater than that laid down in the first paragraph of Article 14, on condition that these vessels pass through the Straits singly, escorted by not more than two destroyers.

The US isn’t a Black Sea power (though I guess maybe if the US transferred the Enterprise to Romania…). Russia can do it because it’s a Black Sea power.

considers

I guess maybe if they got a whole lot of helium balloons and attached them to cables going down to the carrier, they could get the displacement below 15,000 tons.

EDIT: Actually, if they can get enough balloons to offset 80,000 tons, you’d think that they could just do the last 15,000 and convert the Enterprise into an airship and fly it into the Black Sea. The Montreaux Convention didn’t think of that loophole!

Though…hmm. I think that the Enterprise relies on constant seawater cooling for the reactors, so maybe they can’t do that. Maybe the turret does make sense in the context of the helium balloons after all.