The Cape of Escape allows one to successfully pass a move check. In one of the scenarios, you must pass a stealth check or suffer a particular consequence (don't want to post a spoiler!). Is a stealth check a "move" check? In other words, can one discard the cape to pass the stealth check?
Whether we won or lost last night depends on the answer!
Cape of Escape:
"Bury this card to move, or to evade an encounter and then move. You automatically succeed at any check required to move."
With that information, the Cape of Escape does not apply to the situation you have set up.
If you were facing something along the lines of "Characters must pass a Stealth 6 check to leave this location," then the Cape of Escape would let you automatically pass that Stealth check to leave that location.
If you're talking about
Spoiler (click to reveal)
Ripnugget and Stickfoot
Before the enouncer, succeed at at Stealth 9 check or the difficulty to defeat Ripnugget is increases by 2
then no, there's absolutely nothing applicable.
Unless you just use the Cape of Escape to evade the encounter entirely... you know: to escape.