Nice idea!
As far as I know, there is one cycle hire shop in Konstanz - Kulter-Radle (
www.kultur-raedle.de). Not sure if there are others, and not sure if they are ok with returning bikes in Mannheim. Send them an e-mail (kultur-raedle <at> t-online.de) - they should be able to give you more information than I.
Happy riding.
P.S. The owner is quite friendly and speaks good English