I’ve never heard much about what belief systems were common in the region before the Slavs and Khazars—but given the linguistic and cultural connections with Persia, it seems plausible that there might have been some religious influence as well.
For example, could the Bulgars have been exposed to elements of dualism which they then carried west with them, contributing to the rise of the Bogomils and Cathars?
It is possible, but is supported by very little evidence and is mostly supposition. Take for example the following passage found at page 60-61 of the classic work in the study of transmission of dualism/Manichaeism in the west, The Medieval Manichee: A Study of the Christian Dualist Heresy by Steve Runciman.
That an Albanian church of iconoclastic tendencies, disapproving of baptismal and marriage ceremonies, existed in the seventh century is told us by the Albanian chronicler Moses of Kaghankatuk; but he refers to it once only and without much emphasis. If it was an enduring church then it must have flourished in obscurity. But Albania is situated on the north-east of Armenia, where the narrow valleys widen out into the Caspian plain. It looks towards the East. Persian ideas could travel easily there. When triumphant Islam drove Zoroastrianism from the Iranian plateau, it was to such outlying districts that the old religion fled. Albania must have absorbed many Persian Dualists, coming in a thin stream for several centuries; and their descendants would be willing recruits to any Dualist church.
What we get a supposition without evidence (how? what? through whom?):
Persian ideas could travel easily there
That is then quickly elided with a 'factual statement' that has no citation:
When triumphant Islam drove Zoroastrianism from the Iranian plateau, it was to such outlying districts that the old religion fled.
I take this apart because it shows the continued state of historiography around heresy and the theories of 'transmission' of heretical ideas in the middle ages. It is this same kind of supposition that has lead to the widely-held assumption that Cathars in Germany, France and Italy derived their ideas from/were seeded by Bogomils. This theory can be found both in the above The Medieval Manichee: A Study of the Christian Dualist Heresy and then usefully summarized, again with suppositions based on similarity of ideas but not linked in records, in Bernard Hamilton's introduction to Hugh Eteranio: Contra Patarenos (an edition of a tract against Paulicians in 11th c Byzantium)
Going back to Hamilton's forebear: Runciman again relies on typology to try and sort out why certain peoples believed certain things. Here is tries to sort Paulician origins into Bogomilism, and failing an evidentiary basis, roots them back into some broad Gnosticism, without a shred of actual evidence:
We must abandon without result the search for the parent-sect from which [doctrines are given to] the Armenians. John of Otzun declared that the Paulicians thought that they had discovered something great and new. Perhaps they were right; perhaps Constantine, who rechristened himself Silvanus, was a remarkable original thinker who evolved the heresy from his own inspiration. But John of Otzun added that really their teaching was old and obsolete; and this is the more convincing theory. The old and obsolete teaching is the Gnostic teaching, the answer that the Gnostics gave to the problem that tormented them: "Whence came Evil, and in what does it exist?" This answer lay in Dualism. The Paulicians accepted the Dualist tradition from the Gnostics, whether Marcionite or Borborite we cannot know; and by simplifying it, ridding it of much of its Gnostic elaboration, they preserved it, to hand it on themselves to the West.