Transportation from Luxembourg to Bayeux France

All train connections will require you to change stations in Paris - from Paris Est to Paris St. Lazare station by taxi, bus, or Metro. However, there are direct, reserved TGV trains running Luxembourg-Paris several times per day which don't necessitate any additional connections. Book the TGV well in advance for best prices.

Otherwise, take a shorter train ride across the border into France and pick up a rental car near Metz Ville station. You should later be able to return it at any French location with little to no one-way fee. It's handy to have a car in Normandy, any way.

You will have to go via Paris, and change station sin Paris (connect using the métro).
In France, all rail line lead to Paris, so there is not much alternative.

If you want to use a rental car, you would need to take a train across the border (possible to Metz) to avoid the international one-way drop-off fee.