If Emily is suspected as a spy, the interrogation and detaining should fall under the National Intelligence Organization, Turkey's foreign and domestic intelligence agency, not the police.
Turkey or Türkiye, only has a Strategic Agreement with Europol. Therefore, it is almost impossible for them to allow Europol Agent Smitty in. Furthermore, it's far-fetched for her to aid the evacuation of a spy without clearing it with her superiors.
Since it involves a former Marine in a NATO country even with tensions with the US, US military law enforcement should be involved. Example might be the NCIS. The State Department should sent a consular official no matter what.
Emily can not be a Non-Official Cover National Security Agency agent at all as stated. The NSA collects signal intelligence, while it does have overseas stations, it does not send field agents like the CIA.
General Finley changes his role rapidly which is impossible. In The Secrets She Knows (2021), he was from the Pentagon asking the Fly Team to find a DOE staffer. In Crestfallen (2022), he's part retrieving a Russian defector. Now, he wants a NSA staffer evacuated. Throughout, he, a two-star General, also has no oversight of the DOE, foreign intelligence and the NSA.