Does anyone know of a shuttle or other type of direct transportation (Other than renting a car or taking a taxi) .from Honfleur to Bayeux?
I don't think you'll find a direct, scheduled service. PrestoBus and BusVerts each run a few times per day as far as Caen, where you can connect to a train.
According to Rome2Rio, there are two choices of bus companies to take you from Honfleur to Caen, and you can take the train from there. See https://www.rome2rio.com/map/Honfleur/Bayeux-Normandy-France