I started looking at the route with the OSM cycle layer selected on RWGPS and have found 1 water tap icon in some park and a public washroom so far, but I haven't looked at a lot yet.
I found this app,
free taps that is a French based app and hopefully it has more places marked than appeared on OSM. I'll poke around with it and see what shows up.
edit: the app didn't show much along the route, except for some in Brest but I didn't look to see how far off-route they were, so it's pretty much useless. Apparently there are a lot of fountains in Paris