You cannot use the scenario power on summoned Wights (or rather, you can but are always forced to banish the chosen boon).
When you defeat a non-villain bane that has the Undead trait on your turn, choose weapon, spell, armor, or item; display a random card of that type from the box. You may shuffle that bane into a random open location to put that boon under this card; otherwise, banish that bane and the chosen card.
After evading a summoned card or resolving the encounter with it, never put it anywhere other than back in the box unless the card that caused you to summon it instructs you otherwise.
If a card and this rulebook are ever in conflict, the card should be considered correct. There is one exception to this: When the rulebook uses the word “never,” no card can overrule it.
The Traitor's Lodge is not what summoned the Wight, so per the rulebook you are never allowed to shuffle the Wight into a location deck to obtain whatever boon you displayed -- the Wight always goes back to the box. Since the rulebook uses the word "never" here, no cards can overrule it. As such, the displayed boon is always banished due to you being unable to fulfill the requirement in order to obtain it.
However, the Baleful Shadows barrier itself does have the Undead trait (and barriers are banes). If you defeat the barrier itself, you can shuffle it into a random location deck per the scenario power in order to obtain the displayed boon. If you opt not to, the barrier is banished, as is standard for defeating a bane. If the barrier is undefeated, you can never use the scenario power on the summoned Wight.
Nearly this exact question was asked before, although skizzerz's answer is more thorough.
First time I've seen that avatar, skizzerz. Cool.
Yeah, Robot Chris added a bit over 700 new avatars a few days ago, this was one of them :)
Only question now is would it look better with a grey border or purple border? The old one definitely looked better with grey, this one I'm not sure about.