Faolyn
(she/her)
No, it's not at stake because they are in a location where blood-catching vessels will be readily available and visible.Huh?
The player wants a vessel to catch the precious blood that is flowing away. That's something at stake.
Recently I actually saw a version of this scene in the TV show Merlin. One of the character has been poisoned or drugged, and has to be revived by being given the antidote. But the antidote gets spilled. And so one of the aiding characters - I can't remember now if it was Gwen or Merlin - looks around for something to sponge up the spilled fluid and then squeeze a few drops of it into the mouth of the poisoned/drugged character.
That moment of looking around is - in the show - a moment where the tension rises and we, the audience wonder whether something will be found, or whether some other solution will be necessary.
In my session of Burning Wheel, that moment of looking around - at the table, it arises because the player asks, earnestly, "Is there a vessel I can catch the blood in?" - is the moment of tension. The moment of stakes. The moment when the dice get rolled.
The PC isn't looking for a specific vessel or type of vessel (e.g., the naga didn't say it must be presented to it in a silver wineglass or a blessed cup take from the local church). The PC isn't looking for a vessel that has been hidden away, deliberately or accidentally. They're looking for any type of vessel that can hold liquid in a sick room that should logically have cups, bowls, jars, dishes, pots, and other such items. Heck, he could have yanked off one of the decapitated NPC's shoes and had the blood drain into that. And if the PC knew ahead of time he'd be called upon to gather blood, then there should be no reason he wouldn't prepare himself for the task ahead of time unless the game puts artificial constraints on PC actions in order to drive up drama, in which case allow me to link to the TV Tropes page "Idiot Ball."
And that's something I consider to be a bad rule for what I assume is a serious or semi--serious game.
What's actually at stake is the PC's ability to get the blood to the naga, not find a cup.
From what I have read, Burning Wheel doesn't use clocks as per PbtA and many other games. I looked it up and saw a reddit thread wherein some people said "sure, you can import them" and other people said "no, they go against BW's purpose." I don't know what your stance on them is, but you have said that rounds as unit-of-time aren't a thing. And there was no apparent opposition that was keeping the PC from catching the blood--there was an assassin, but no mention of the assassin attacking the PC, at least none that I can remember.
Which means that, from everything you have said, there was no actual in-game or character-driven pressure on the PC to obtain a cup, nor was there a time crunch. Any pressure in the scene was from GM fiat.
So again, what's actually at stake, and what's actually interesting, is the PC's ability to get the blood to the naga.
In other words: to get around residents, guards, servants, visitors, and various other onlookers who may have just learned there was an assassin in the house and thus will be on high alert and/or freaking out, while carrying a cup full of blood, and possibly even covered in blood from attempts to get the blood into the cup.