I doubt it will make that much difference. The amount of non-alcohol non-water stuff in rum is fairly trivial for the level of chemistry we're talking about. Maybe if I was using a dark Jamacan rum, too, that might be significant. But this is just Bacardi, and chemically, I suspect that the difference between Bacardi and vodka is basically a few ppm of aromatic esters.

. . .

Okay, I just had a teaspoon of Bacardi while holding my nose, and there IS a very slight taste difference between Bacardi and vodka, but not a hell of a lot. Without aroma, there's not that much rum-like about Bacardi. It's all in the aroma. So I continue in my suspicion that there won't be that much difference for this level of chemistry.