Apologies for my ignorance.

Army of the Faithful (Baratheon) is challenging my Targaryan opponent in a military challenge.

My opponent only had two characters in play: Daenerys (who is already kneeling) and Strong Belwas who isn't.

Army of the Faithful has strength 6.

My opponent has two gold in his treasury and no way to win the challenge, meaning he has to kill off 2 of his characters (since the claim value on my Baratheon revealed plot card is 2).

But, Strong Belwas says 'Pay 2 gold to save a noble character from being killed' (i.e Daenerys).

The question is: Can Strong Belwas save Daenerys before he has to die himself? Or do they both have to die at the same instant (meaning he can't save her first)?

Saves work as interrupts. That is, the kill effect initiates, the save effect is played, and THEN the kill effect resolves (unsuccessfully for the saved character).

So, even though Dany and Belwas are chosen to die at the same time for the same 2-claim effect, the save happens before the kill resolves; meaning that neither one is dead when Belwas saves Dany.

That's the basic answer on the timing of saves. There is a deeper, more technical answer about how it doesn't actually matter because the advanced timing of this game does allow for Responses, including saves, to be used "on the way out," but none of those advanced, technical rules apply to the situation you describe. Belwas isn't dead when it's time to save Dany.

Thank you for answering! I assume you mean "(unsuccessfully for the character who is doing the saving)" i.e. Belwas?

No. If the kill effect resolves successfully, the character dies, right? So in your scenario, the kill effect (claim) initiates, Belwas saves Dany, then the kill effect resolves -- successfully on Belwas (he dies because nothing saved him) and unsuccessfully on Dani (she doesn't die because Belwas saved her).