By Lexi Carson Associate Editor [This story contains spoilers from Love Story, episode five, “Battery Park.”] While Love Story: John F. Kennedy Jr. & Carolyn Bessette is, at its core, a romance, it doesn’t shy away from the couple’s most painful moments. Thursday night’s episode, “Battery Park,” revisits one of their most infamous: the heated 1996 argument after they left their Tribeca apartment at 20 N. Moore St. and walked to Battery Park. Though the time period was before smartphones and viral TikTok videos, paparazzi captured the fight on camera, and the footage was featured on tabloid front pages. In Love Story, the episode imagines what may have led to that very public unraveling. Over the Fourth of July weekend in 1995, John (Paul Anthony Kelly) proposes to Carolyn (Sarah Pidgeon) during a fishing trip on Martha’s Vineyard. She doesn’t immediately accept. Unsure whether she’s prepared to become Mrs. JFK Jr., Carolyn asks for time.…