I'm a bit unsure about the timing of the card feat at the end of an adventure.
I assume each adventure gives you a card feat, given what we've seen so far.
This means that each person can rebuild their deck at the end of the adventure.
But when exactly do you "finish" the adventure? Right after the scenario when you rebuild the decks, or is the adventure finished at a time after that and you have to discard loot, etc?
The rulebook doesn't seem to be clear on this. It tells you what to do when you finish the scenario, but doesn't tell you whether the scenario finish is the same as adventure finish.
This matters quite a bit because you may want some card from the last scenario in addition to everything else when you gain the card feat. Or you may not want the card because it's unusable by you (eg Merisiel gaining the weapon card feat, but the only ones available from loot in the scenario needs proficiency, and she doesn't have it yet.)
I assume you finish the adventure the moment you finish the last scenario in the adventure, and so when you take your card feat you may fill that slot in your deck by selecting from the cards you and your companions have acquired.. And even if it is unusable you must rebuild from what is available.
I don't think being forced to go find a basic card makes much sense.
So to be clear, the order would be:
1. Finish scenario, gain reward.
2. Finish adventure, gain reward.
3. Rebuild deck as per normal rules.
There's no such thing as an "unusable" card in this game. No card "requires" proficiency, so if you don't have it you can still use it at some penalty (same applies to spells, etc).
It may not be your choice pick, you can still use it. Grab the one that works best for you and trade it in later when you find something better.
Oops. Sorry. I didn't mean unusable, I meant undesirable. Copied the term from the OP. So to re-affirm the rule, you only get a card from the box when rebuilding if there are no cards left over from what you and your teammates acquired. And you get a basic card if you have to do that.
So, if you are Ezren and all the items you had were potions and they all got banished during the game and you are sitting there rebuilding your deck, if your teammate Sajan is not going to keep his Amulet of Might Fists, then you are going to have to take it. No matter how useless you might think it is to Ezren.
But if your teammates have no items they won't be keeping, you can search through the items in the box and pick from the basic ones to complete rebuilding your deck.
So, bringing this full circle, when you get that card feat, you have to fill it in with what's on the table. If, and only if, nothing is available you can go find a basic card to complete your deck.
And, to reiterate, I think you got that card feat as soon as the last scenario ended, so you rebuild your deck with one more card.