To prepare ice cream, combine first 3 ingredients in a saucepan; bring to a boil. Stir to dissolve sugar; cook 3 minutes.
While sugar syrup cooks, place egg whites in a large bowl; beat with a mixer at high speed until foamy. Add cream of tartar; beat at high speed until stiff peaks form. Gradually pour hot sugar syrup in a thin stream over egg whites, beating at high speed until mixture is glossy and stiff (about 5 minutes).
Combine buttermilk, cream, 1/3 cup juice, and 1 teaspoon rind, stirring with a whisk; gently fold into egg white mixture. Spoon into a freezer-safe bowl; cover and freeze 8 hours or overnight.
To prepare pineapple, pour 1 cup of boiling water over 2 tablespoons rind in a small bowl; let stand 5 minutes. Drain. Repeat procedure with remaining 1 cup boiling water and rind.
Combine 1/2 cup sugar and 1/2 cup water in a small saucepan. Bring to a boil over medium-high heat, stirring constantly. Cook 3 minutes or until slightly thick (do not stir). Remove from heat; stir in 2 tablespoons rind, 1/4 cup juice, and chile. Cool completely.
Place pineapple slices on a platter. Pour chile-lime syrup over pineapple; cover and chill at least 30 minutes. Serve pineapple and syrup with the ice cream.