If I was you (DFR) it might be more interesting to first do the JAA in the US. There are a few threads on this site about doing the JAA training in the Florida; when I was at EFT there were a few French fellows like you as well. Some students came in for a JAA/FAA combo, but if you do it all in one go it is intensive and I doubt you can do it in 5 weeks unless the weather is exceptionally good, the school not too busy and you are well prepared.
I also did notice that those who passed the JAA with flying colours were not that good at the FAA exam; the main reason I think behind this is the intensity of the previous weeks and the induced self-confidence after passing the JAA.
In my opinion getting the FAA as a standalone at a later stage is actually a good kind of refresher exercise, but of course there are extra expenses. The FAA exams are different, at least in the formulation.
On another note, what is the benefit to have a FAA PPL as a standalone? You can also apply for one based on your JAA PPL i.e. ; and should this document lapse for any reason other than a purely administrative one then I really wonder why the licences issued by other states should not be considered void as well.