The Haeberlins who run the Auberge de l'Ill are really nice down-to-earth folk, and whilst it may not be the most frequently asked question, I think they would not be offended if you asked if they could look after your cases during your meal. In addition, the same family run the Hôtel des Berges at the other side of the restaurant garden, and it might be possible for you to keep the cases at reception of the Hotel des Berges during your meal if the cloakroom arrangements at the Auberge de l'Ill are too restrictive.
Of course you could always spend the night at the Hôtel des Berges, and REALLY push the boat out!