Make the bell pepper juice: Puree the bell pepper in a high-powered blender (or juice in a juicer). Strain through a fine-mesh sieve.
For each cocktail: Combine 1 ounce bell pepper juice, the rum, tequila, lemon juice, simple syrup, caraway liqueur and bitters in a shaker with ice; shake for 30 seconds. Strain into a coupe glass. Garnish with lemon peel.