Buy a set of these and ride off into the sunset.

https://www.velomine.com/index.php?m...roducts_id=739

They also come in 700c and CR-13 skinny 700c versions. I've easily got over a 1000 miles on a 700c set. depending how "French" this bike is you might have to buy a new freewheel, which are available in 7 speeds. Yes there are 120mm 5 speed free hub rear hubs available but pricy.

https://www.hubjub.co.uk/sunxcd-120-...hub-5204-p.asp

I would assume that the original derailleur will shift across as many speeds as you want, but that gear is not particularly durable, so If your going to put miles on this bike you might want to swap out for some more durable components, and save all the original parts for a later restoration for display.