Negotiators for Britain and the European Union on Tuesday hammered out a draft Brexit deal, setting up a critical endgame in which Prime Minister Theresa May must now try to sell the plan to her skeptical ministers and Parliament.

Following more than two years of wrangling over Brexit, Mrs. May is set to put the proposed pact before her deeply divided cabinet on Wednesday. Her political future and that of her Conservative Party could well depend on the outcome. If the prime minister doesn’t get the necessary backing, she could...