Oleksandr,
I think this might be the best choice to pursue, but I am not sure if they will be close to your location in Lyon (see address below):
Chovin Roland
Description:
Motorcycle, Scooter and cycle hire shop located at 41 avenue Louis Masson, Chabeuil (26).
Contact Details:
Tel: 04 75 59 25 53
International: +33 4 75 59 25 53
There is also this firm, but they seem to specialize in guided tours rather than just rentals, but might be willing to let a bike for rent:
Orientation Velo
Description:
Cycle hire company with experienced instructor and guide for day tours in and around the city. Based at 212 avenue Felix Faure, Lyon (69).
Contact Details:
Send an e-mail to Orientation Velo
Tel: 06 50 53 88 37
International: +33 6 50 53 88 37
Website:
http://www.orientation-velo.com/This firm offer electrically-assisted bikes (not what you are seeking, I suspect).
Bike and See
9 place Antonin Poncet (Bellecour) – 69002 Lyon
Tel. +33(0)4 72 41 84 80
www.bike-and-see.frElectrically assisted bikes with multimedia guide in English.
http://www.bikerentalfrance.com/ Claim to offer carbon road bikes in the Lyon area, but they are very expensive to rent and do not come with pedals (!).
Might it be possible to take a taxi, metro, or bus to the bike-hire station?
Best,
Dan.