There is a hefty fee for picking up a car in one country and dropping it off in another. If you rent a car, I suggest that you take the train from Bruges to Lille and pick up a car there. It's 90 minutes from Bruges to Lille by train. According to viamichelin.com, it's about a four-hour drive from Lille to Bayeux.
However, according to the timetables at www.bahn.de, it's possible to take the train all the way in about the same amount of time. There is one Thalys train that stops in Bruges. Take it directly from Bruges to Gare du Nord, leaving at 06:26 and arriving at 09:05. Either take the RER E from Magenta to Haussmann--St-Lazare or take a taxi from Gare du Nord to Gare St-Lazare. Leave Gare St-Lazare at 10:10 and arrive in Bayeux at 12:15.