Briancon would be good for the road (or Vanosc/Bourg d’oisans) but XC might be a bit limited. Having said that, there are a few waymarked trails going out from Briancon towards Montgenevre.
Pyranees might give more scope for a Xc and road combo, somewhere close to Luchon or Lourdes (tourmelet etc).
Thinking we might fly up to paris to see the champs elysse tdf finish while we are there.
I was at a finish years ago when my sister lived in Paris, it was an occasion but I wouldn’t break up a trip to the alps to do it again. Really busy, difficult to see anything and not very interesting to watch. If it was me, I would miss out on the logistical hassle and watch it on the tele.