Use a vegetable peeler to remove the green lime peel (strips), taking care to avoid the white pith. Put the peels in a large bowl. Juice the limes and set aside the juice.
Add the strawberries and juiced limes to the bowl with the lime peels. Add the sugar and crush the fruit. Let stand at room temperature, stirring occasionally, until sugar is completely dissolved, about 1 hour. Pour through a fine sieve and discard solids. You will have about 1 3/4 cups of strawberry mixture. Store in an airtight container in the refrigerator for 1 week.
To make 1 drink, combine a shot of white rum, 2 tablespoons lime juice and 3 tablespoons strawberry mixture in a cocktail shaker. Fill the shaker with ice; cover and shake until everything is well chilled and mixed. Strain the drink into a glass and garnish with a strawberry and lime wheel.