Actually, this is a bit of a funny one.

The train you will be on appears to be a hybrid - CNL 479 (reservation only) from Hamburg to Zuerich HB, which sets down only from Karlsruhe onwards; and IC 61419, from Duisburg to Basel SBB, which picks up and sets down at each station, and doesn't require a reservation.

The two trains run together from Mannheim to Basel SBB.

Consequently, you can buy a ticket for the Basel Bad - Basel SBB section for this train on the DB website, and just stay on the train. (If the CNL conductor gets bolshie - unlikely - just move through to the IC portion of the train at Basel Bad ).