I strongly suspect all aspects of their dedicated air cargo operation will remain contracted out so callsigns will remain as-assigned to the indivdual operators of the actual aircraft. ("We bought/leased/branded these airplanes and we'll pay you to fly them for us.") I doubt The House of Bezos wants to get into the complexities of aircraft and crew licensing, flight planning, crew scheduling, flight training and repair/overhaul/maintenance/engineering/certification of the aircraft. At some point in the future perhaps they will purchase outright or at least have controlling interest in an actual licensed commercial freight operator.
FYI, callsign telephony is assigned to the operator as published in ICAO document 8585 "Designators for Aircraft Operating Agencies, Aeronautical Authorities and Services" so they will follow the aircraft operators designation.