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.
That's not true. You CAN unlock sets in the collection with pieces from reputation vendors, but you need to wait until the refund timer has finished and the item gets permanently bound to your character.
I unlocked the Thana Vesh set this way, by combining the chestpiece from rep vendor with the rest of the set bought from GTN.
The only items you can't unlock, are those which are exclusively available on the rep vendors, e.g. the golden Triumvirate set.