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 untrue. If items are not being added to collections through the ICE vendors, then that's a change in the way these vendors work. Through the Bounty Supply vendors, I purchased the Mira's Armor set and the Miniature Isotope-5 Droid and they added to my collections. Through the Contraband Resale vendors, I purchased the Diplomat's Hoverchair, Ruddyscale Kowakian Monkey-Lizard, and Shadow Nekarr Cat and they all added to my collections. For the armor sets, they would add to collections once the bind timer wears off or if you added mods to accelerate the bind process. Frankly, it SHOULD work this way too as Cartel Certificates come from packs so this stuff is still being purchased through the CM.