Yes, the French Nuclear industry is one of the bigger ones around, however civil nuclear reactors use 3-5% Enriched Uranium, Current French "Caramel" reactors use c. 20% compared to US/UK naval reactors 90%+. France has produced HEU reactors in the past but current knowledge is really on the low end IMO.As the MN is unlikely to ask. The nuclear industry is very powerful in France and having a US reactor in the MN capital ship would be completely unthinkable.
Advantage of the 20% is it can be exported as the low enriched uranium can't be easily turned into "da bomb", disadvantage is it needs refueling more often.
It could be done if you really really wanted to, but in the real world of limited resources do you spend the $1.5bn on a new reactor or on extra planes/ships of a conventional carrier group? Same dilemma as RN with QE powerplant!