The arguments for 2 seem generally absurd. People constantly try to make these "trust" arguments - and frankly history has shown them to almost always be wrong.
Hanyi's just said she can't do it. Taking her at her word and accepting that our little sister needs help, because she's being screwed by factors beyond her control, isn't "not trusting her". Hell, if anything it's trusting her assessment of her own capabilities (though that would also be a weird way to frame it).
And Hanyi will be annoyed? Of course she will be! She's being sabotaged when she thought she was going to have a great triumph! It doesn't mean this is going to super poison our relationship or anything, and being super paranoid about ever making our family upset with us at all is really not the lesson people should be taking from all of our heart demon shit. Indeed, if anything the problems are caused by LQ being unable to deal with her family being upset with her at all, and she needs to learn how to deal with that and work out when it's fine.
The odds on 2 are not good, and frankly failing kind of spikes both our quest and Hanyi's goals here. There are layers upon layers of consequences here.
Really, the choice should be between (1) or (3), with questions around "what is the actual intention with (3)?". Like, people are jumping straight to "beat up a river", but LQ's history and leanings here that it's more likely that she'd try to surreptitiously run interference and personally placate the river imo... Indeed, we actually know it can't be that because we know from our experience that that doesn't work (unless you're the Bai). Killing spirits just means they come back angry later, and flood things, so I don't think it would actually count as "success".
[x] Learn the missing piece yourself and take part in the performance (80% Success. Improved relationship with Clergy on success. Negative relations with clergy and Nobles on Failure. Reduced benefits for Hanyi's reputation. Annoyed Hanyi)
[x] Let Hanyi perform with the current choreography. Make sure this Stream doesn't cause trouble personally. (100% Success, +1 Power Success, ??? consequences. Reduced Clergy Reputation.)