Mission accomplished, I think. I found a nice hotel in Modane, and also determined (via emailed question) that they provide secure bicycle storage. It's the Hôtel le Commerce, and looks nice on the inside (based on Google Maps review photos) and has reasonable prices. Unfortunately, I won't be able to return there until 2022, but it will make a good base for cycling one of the best cycling valleys in the world, the Vallée de la Maurienne
There are so many good climbs going up from that valley, not surprising really, given the valley is 110km long
Guess where Modane is located in the valley ? In the exact middle, that's where! So it's ideal for exploring as much of the valley as possible.