Even if the rehearsal dinner location doesn't allow food, they may allow a cake, so ask (they often charge a "cake cutting fee"). If you cannot have the cake at the rehearsal dinner, you can have it at the wedding. They'll put the groom's cake next to the wedding cake and guests will get a choice which cake they want.
Traditionally, groom's cakes were served at the wedding. Years ago, you could only get a wedding cake in one flavor. To give guests options, brides (initially in the US South) introduced a second cake. The bride choose the wedding cake flavor, and the groom chose a different flavor for the second cake (hence "groom's cake"). Now, in most wedding cakes you can get each layer as a separate flavor, basically making the grooms cake more of a tradition than a functional component of a wedding.