The third party is for the vote, not the treaty-signing (although having the third party present for that may be a good idea).
As I see it, the first order of business is to come to an agreement with Remilia on who the third party will be. We'll try for Suika first, then ask about having Satori do it if Remilia says no to Suika (and the Dragon agreed to that plan, so don't be trying to tell me that's not the right way to go about it). After that, we'll go to Suika/Satori and ask for her cooperation. If we end up needing to ask Satori, we can find Nue while we're underground; if it's Suika, we'll return to Remilia first. Either way, we should find Nue. Yes, the Dragon said that giving Shuuei a truly perfect disguise would only be a temporary solution, but that's all it needs to be (the idea is that Shuuei will get less uncomfortable of interacting as she does it more, eventually going from Nue-assisted disguise to the outfit from Yukari to, hopefully, no disguise at all).
>What do we know about the events of Undefined Fantastic Object, especially in regards to why the vault fragtments looked like UFOs?