The Prime Minister of which one of the following countries is chosen by the ruling prince from a slate of three candidates put up to him by the President of France ?

The description fits Monaco. Under Monaco’s constitutional practice the head of government (Minister of State) is appointed by the Prince but is chosen from a shortlist of three candidates put forward by the French authorities. Freedom House and other sources state that the Minister of State is traditionally selected by the monarch from a list of three French nationals submitted by the French government, and official summaries note that the Prince appoints the Minister of State from three candidates proposed by the President of France, reflecting the special Franco‑Monégasque arrangements concerning senior appointments [2]. This procedure distinguishes Monaco from San Marino, Liechtenstein and Malta.

Sources

1. [1] https://freedomhouse.org/country/monaco/freedom-world/2021
2. [2] https://2009-2017.state.gov/j/drl/rls/hrrpt/1999/348.htm