If my experience of going abroad as a student and looking for a bike is anything to go by, people will be falling over their own feet to help you out with a used bike. You will then be offered one absolute heap of unwanted scrap iron after another and folk will be insulted when you politely decline.

Dragging through 'Le Craigs Liste' or whatever they have there sounds pretty dire too, and a big risk of getting a stolen vélo.

Lyon must have plenty bike shops, I'd wander around for a couple of weekends until I found the right bike. Do agree you'll want one early in your stay. Invest in a good lock too.

Suspension shouldn't be a deal breaker for touring in Europe, a normal folder is fine for anything down to farm tracks and footpaths. You will need low gears if planning to follow any coasts.