There were many different Emperor's who vacillated on using the Papacy as a counter against the influence of various Patriarchs of Constantinople. It's no coincidence they were usually in precarious political positions like Michael VII (and Michael VIII), but it doesn't change that vacillation or that the Papacy was a useful counterweight against the insolence of the likes of Cerularios.
A Bulgarian desiring legitimacy but would be a different position, and there's any... angst that might exist from the various campaigns of Basil and the legacy of the Tsars. Sure, they can assimilate to the Greek rite, as well, but if there's a remnant of significant political resistance that's using that rite, it's could well be more useful to be religiously distinct.