Which "hat" gives the president the power to negotiate treaties?

The right answer is the Chief Diplomat Hat

The United States Constitution allows the president to negotiate treaties with other nations. As the country’s Chief Diplomat, the president has the power to oversee the nation’s foreign relations as well as appoint the country’s representatives to other nations.

 The hat of CHIEF DIPLOMAT

Further details:

The various "hats" worn by the President, according to an article by Scholastic on "Seven Roles for One President," are these:

In regard to the President's role as Chief Diplomat, Article II, Section 2 of the Constitution gives him that authority -- with the advice and consent of the Senate.  That section of the Constitution says of the President, "He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur."

The advice and consent of the Senate typically comes after the treaty has been negotiated by the President.  As reported by Justia (a legal information site), "The Senate’s role in relation to treaties is essentially legislative in character."  In a Supreme Court decision in 1936, this was affirmed by the Court's decision, which said, that the President "alone negotiates. Into the field of negotiation, the Senate cannot intrude" (United States v. Curtiss-Wright Export Corp., 1936).   However, for the treaty to become active, the President must secure approval by the Senate, with a two-thirds majority vote, as stipulated by the Constitution.

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