You also could have used the word "Barding" with the partial match option highlighted and gotten them to pop up that way(i believe the only one that would miss is the "Egg Harness", which being from a past live event you are unlikely to ever see for sale at a reasonable price anyway).

As for ranking up within your GC, the first handful of ranks are just purchased with GC seals(and if youve already cleared the GC hunting log, then those requirements will already be met when you get to them). Once you get through those the next few ranks come with added quests. These quests involve clearing out an optional(non-MSQ) dungeon. Then the final(current) rank comes from forming and leveling up your squadron. But you can buy all 3 company bardings well before that point.