Cachaça (not slickly speaking a rum) adds distinctive grassy notes to this three 'rum' Daiquiri. Try swapping out the cachaça for rhum agricole and indeed...
works both with and without the bitters. It's difficult to get the sugar to dissolve if you do grind it in a mortar and pestle so if you want the full effect mix early and stir in the sugar and leave to sit.