You're on the right path here. The items from reputation vendors do not unlock the piece in Collections. If you were to compile the set from different, non-reputation sources (such as some from the Cartel Market, some from the GTN), it would unlock. Reputation items, however, are specific to the character with the rep.
Is the pieces from the ICE vendor really compatible With the others? Guess you will have to buy all the pieces from the GTN not just 4.