After you buy the set equip the pieces that way it is bound to you and your alt of the same armor type of either faction can use it for tmog.
I first equipped the armor on all characters and that didn’t work despite “learning the appearance.” I went through a bunch of workarounds I listed above since since equipping didn’t work. It solved most of the issues except cloaks. I still think something is broken with it considering I had a cloak appearance I had not yet learned before, but not the one I did lol.
I’m running into the same issue with having purchased Argent Conscript gear, equipping it, but it’s not appearing properly in my transmog appearances tab. I logged back into the original toon that purchased & equipped the items just to make sure I hadn’t actually forgotten some of the pieces of gear, but that toon shows all of the gear slots properly on the appearances tab. Something’s definitely not working properly. This is happening with all the different armor types, only showing up fully on the toon that bought them, but missing pieces on my alt’s appearances tab.