Combine the ginger and lemon juice in a food processor and process until minced, stopping the machine periodically and scraping down the sides, if necessary.
Place the puree in Instant Pot with the natural sweetener and 1-quart water. Add the lemon peel to the pot. Cook on high 30 minutes. Slowly release pressure. Cool slightly, then strain and chill.
To serve, place about 2 tablespoons of the ginger mixture in a glass full of ice. fill with carbonated water; taste and add more ginger mix if you like. Garnish with a lime wedge, then serve.