We discovered the use of rum in place of cognac in a Vieux Carré thanks to a comment by Collin Westgate on our Vieux Carré page saying that he'd discovered...
This is a great variation, but I find it's much nicer if you substitute the Jamaican rum for Rhum Barbancourt 5 star and replace the Rosso vermouth with Lillet blanc. The barbancourt falls closer to cognac in flavor so it reads a bit more like a vieux carre, and the Lillet blanc adds a needed touch of bright orange character to an otherwise slightly heavy drink. Somewhere between a vieux carre and an el presidente this way, and is even better imo with an absinthe spritz as garnish.