Mixing certain cocktails with a given spirit has a reason to be, hence why today it's possible to study mixology, which provide people with the knowledge needed for those cocktail recipes we can find here an there.
In the case of Bloody Mary, the choice of vodka isn't a caprice someone had, but the fact of being a flavorless spirit, what contributes to preserve the taste of the other ingredients without variation other than the mix itself.
When combined with whiskey, irremediably its taste will change, and depends on what kind of whiskey you use to distort the taste (and color) of the original shot in minor o major degree.
Personally I have never tried a Bloody Mary with whiskey, but I have tried it with Gin, that being similar to vodka, its taste to juniper berries give this shot a special note.