Appleton White is a good choice, it has some character and is not too bland. Still, next to Wray & Nephew's Overproof Rum it is quite different. The mint and lime in Wade's recipe would perfectly match the strong unrefined taste of the Overproof (although its feisty taste is not for all to be sure). I have to remember that Cuban rums are not available in the U.S. But Wray & Nephew is a good alternative and the young unrefined rums around the Caribbean are not all that different. I like dark and golden rums but for this drink a white rum seems to me essential although it would work probably too with a dark rum (after all, think of the mint julep).