It is about 7 1/2 to 8 hours to get from Frankfurt airport to Bordeaux, with 2 train changes including a change of stations in Paris, from Est to Montparnasse.
If I am understanding this correctly, you will have a separate ticket from FRA to Bordeaux if flying. I don't know the schedule offhand, but that would be about 2 hours to check in again at the gate, go through security again, get to the new gate and board the plane, then about 2 hours in the air, and another half hour to deplane and get your luggage. So that makes it 4 1/2 hours.
What does Luxembourg have to do with it? You still have to get from FRA to Luxembourg, and trains from Luxembourg also go to Paris Est, so what is the difference, other than a longer and more tortuous journey?
If you are not already in Frankfurt, why not buy tickets from your home airport to Bordeaux?
PS, Sorry, I did not go through all your previous posts, but the itinerary seems to have significantly changed from this one. https://community.ricksteves.com/travel-forum/general-europe/mother-daughter-trip-across-4-countries-technically-5