Unless they're planning a second stage of the coup that would be better to cut off, or if part of the point was to corrupt/mellow out Bhava-Agra by dumping it in the middle of Gensokyo. Likewise, I doubt there's enough peaches to keep people pacified for very long, and the Incident Solvers are going to end up pummeling SOMEONE for this, and Iku's on the list.
However, I think that if we can't get a good track on Tenshi in short order, it would be better to take our time a little, since the trail would have grown too cold to gain much from confronting her as quickly as possible. That point may have passed already after doubling back to speak with the outworlders again - I made the 'tracking' suggestion more out of pedantry and pondering loopholes than as a serious idea. At this point it might be a better idea to at least spend time working out some of the problems on the islands while working on finding where the hell Tenshi got off to and what to do about whoever's responsible for this, which I'm guessing was your point anyway.