The Talisman is tagged as a "Enchanted Heirloom" and is not classified as a Relic.
Villian side:
Enchanted Heirloom (tag)
When flipped to this side, restore this card to 7 HP and put it in the villain play area.
This card is indestuctible.
When reduced to 0 or fewer HP, flip this card.
Increase damage dealt by Kismet by X, where X = the number of jinx cards in play.
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I translate this in terms of End of Days as the card remaining unharmed. End of Days *destroys* cards and the Talisman is immune to this effect, thus short circuiting the destruction cycle - EoD proceeds to another target. So it would remain where it was (villain or hero side) with however many HP it currently has.
Now, interesting enough, I think playing End of Days while Lady Luck was in play *could* affect the Talisman as a secondary effect.
1. End of Days attempts to destroy Lady Luck (or actually, any of Kismet's cards that are in play)
Lady Luck (On-going, Lucky)
Whenever a villain could would be destroyed, reveal the top card of the villain deck. If it is not a Lucky card, discard it and prevent the card destruction. If it is a Lucky card, put it into play.
2. The top villain card is drawn - it is "Fortune's Smile" - a One-shot Lucky card. It is activated and put in play per the Lady Luck card.
Fortune's Smile (One-shot, Lucky)
Kismet regains (H) HP and deals each hero target 1 Psychic damage.
Play the top card of the Villain Deck.
3. Fortune's Smile triggers the next card in the Villain Deck - which can be "Unlucky Break" and it will attack the Talisman if it is currently in front of a Hero, potentially "Killing" it and causing it to flip.
The sub-processes complete with Lucky Break, Fortune's Smile, and Lady Luck going to the trash. End of Days can now proceed with scheduled destruction. Though any attempt to destroy Lady Luck with a on-going destruction card can trigger this same sequence.
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I can see this going either way.
a. End of Days is "persistent" in destroying non-relics that can be destroyed and will continue going after Lady Luck until it "gets through". Which does leave the players with the option of saving it for last and trying to burn through Kismet's deck - with all the Lucky one-shots getting triggered! Or go for it first in order to attempt to minimize the damage, and potentially failing to have it work out well.
b. End of Days only gets one chance to take out a card and if it fails it moves on to the next. Leading to the situation of End of Days generating the big glare of bright light as it takes effect and wipes the slate. And the glare fades to slow Kismet and her array still there and intact (with a pile of non-Lucky cards added to the trash) going "Nyah Nyah" at Fanatic.