If I'm holding a Battlefield and have a Pack of Wonders ready, on the opponent's turn they initiate a Showdown, can I exhaust my Pack of Wonders to send my unit to my hand instead of dying?

No, you cannot use Pack of Wonders to save your unit in this scenario.

The Ruling

You cannot activate the ability of Pack of Wonders during a showdown.

According to Rule 145.2: "The Activated Ability of Units may be executed at any time during the controlling player's Main Phase during an Open State, and not during a Showdown." While this rule specifically mentions "Units," RiftJudge FAQ #8278 clarifies that this restriction applies to all activated abilities (including those on Gear like Pack of Wonders): "Regular activated abilities (without Action or Reaction keywords) cannot be used during combat/showdown, regardless of whether the state is open or it's your turn."

Why it doesn't work:

1. Timing Restriction: Pack of Wonders has a standard activated ability (indicated by the [E] cost). It lacks the [Action] or [Reaction] keywords. Therefore, it can only be activated during your own Main Phase in a Neutral Open state.
2. Showdown State: When your opponent initiates a showdown, the game enters a Showdown State. During this state, you are restricted to playing [Reaction] cards or [Action] spells (if you have focus). Because Pack of Wonders is neither, its ability cannot be activated.

Summary

Because Pack of Wonders does not have the [Reaction] keyword, it cannot be used to respond to the showdown or the combat damage step. You must have activated it during your own Main Phase prior to the showdown to return your unit to your hand.