Bring figs, water, corn syrup, sugar, heavy cream, ¼ cup Port, and salt to simmer in small saucepan over medium heat; simmer, stirring frequently, until the liquid thickens and figs are softened, about 8 minutes. Transfer fig mixture to blender and process until smooth, about 1 minute. Using bottom of ladle and pushing in circular motion, strain puree through fine-mesh strainer into bowl, pressing on the solids to extract as much puree as possible (you should have about 1 ½ cups puree). Discard the strained solids. Whisk butter, remaining 1 tablespoon Port, and vanilla into the fig puree.