I know fleur de lis was born in 6th century Frankish Kingdom and used on several coat of arms. (Naples, Tuscany etc.) But these countries have connections with actual Franks and then French one way or another but I can't help but imagine what kind of relation is there between French and Bosniaks in Middle Ages? Maybe from Charlemagne's empire?
There is a French connection here, but from much latter period.
Coat of arms of the Kingdom of Bosnia is in fact coat of arm of its ruling dynasty- House of Kotromanić. Until Tvrtko I declared himself king in 1377, House of Kotromanić ruled Bosnia as vassals of the king of Hungary. And Hungary itself from 1308 till 1386 was ruled by the House of Anjou- a cadet branch of Kapet dynasty.
In one document dated 1432 otherwise unknown founder of the dynasty- Cotromano Gotto is called relative of the Hungarian king. He should have leaved long before Anjou become kings of Hungary, but earliest versions of the Kotromanić arms had red instead of now better known blue (azure) and thus clearly derived from Arpad red-white-stripes.
It should be also mentioned, that armorials provide a number or others coat of arms for Kotromanić's. For example Wernigeroder Wappenbuch (c. 1486–1492) and Ulrich von Richental's Chronicle of the Council of Constance gives white stripe on red quartered with crowned man holding a sword. Armouries from the latter period, like Korjenić-Neorić Armorial(1595) and its derivative Fojnica Armorial (mid 17th century) provide entirely different coats for Bosnia ( C shield in Fojnica Armorials (from 17th century) combined arms, copied from Korjenić-Neorić), while House of Kotromanić are given a lion, most likely deriving from the coinage of the Tvrtko I.Fleur de lis version do appear in the Korjenić-Neorić (and Fojnica) armorial, but as arms of Tvartkovic (descendants Tvrtko I),that is the royal branch of the House of Kotromanić.
