Is it even possible (at least 1/10) to beat "A knife in the dark" scenario with just 1 core and the 1 st saga expansion?

We struggled through it in over 4 hours with 3 players, using Gondor, Silvan and Ent decks. Two of the decks were defeated and the third one (ents) managed to win with one remaining hero and 48 threat. And we use all packs and spend quite some time deckbuilding. So if it is possible, it's going to be very, very difficult.

But those decks don't exist with just a core set and Black Riders.

I don't think you can get a 10% win rate with just those boxes. Assuming you are playing 2 handed or 2 players, your best bet would be Merry, Sam, and Pippin paired with probably Legolas, Eowyn, and Aragorn or Beravor. At the very least you might want to get The Dead Marshes so you can get Fast Hitch for your Hobbits.

You can also play on easy mode. This situation is pretty much exactly what it was made for.

But those decks don't exist with just a core set and Black Riders.

No, that was my exact point. Even with a complete card pool, AKitD can be a very difficult quest. Easy Mode is probably ynaca's best bet.

Easy mode is indeed highly reccomended!

I played the Black Riders box with a fairly limited card pool at the time (single core + mirkwood + KD and first two packs from Dwarrowdelf). This is more than you have available, but most of the extras you get aren't that useful for the Hobbit deck I'd reccomend using there.

A knife in the dark is hard in two ways: the cards are tough by themselves, and you need to know what the quest is doing. Most importantly, you need a strategy for Bill Ferny. From your cardpool, Son of Arnor is the best bet to get him out of staging. Once you're rid of him, quest at a decent pace and prepare for the final encounter. Unlike Shadow of the past, this quest encourages building up some allies.

Yes,I made it a lot of times.

Some advices:

1.Do not let Ringwraiths into Encounter Deck.

2.Use Tactics Merry+Daggers of Westerness/Dunhere+Daggers of Westerness.

3.Use Son of Arnor.

4.Use threat reduction.

5.Always commit at least 2 characters to quest.