The Faceless Men could be looking for ways to destroy the dragons.
Though the fact that "Pate" is a FM is obvious, that never made much sense to me. The Faceless Men as an order predate Braavos. Braavos was founded 800 years ago so the Faceless Men are older, possibly much older than that, since they started in the Valyrian mines.
Aegon's Conquest happened about 300 years or so before the series, so the Maesters have only been in contact with dragons for about 200 years(it's been about a century since the last dragon died).
Now why would the Faceless Men need dragon lore from the Maesters if they(the FM) have been founded in Valyria itself and have been around dragons much longer than the Maesters? And why now, since at the time Braavos was founded the Valyrian Freehold was still alive and well? Or why didn't they look for it while the Targaryens still had dragons? Surely Aegon's dragons were far bigger threats than Dany's baby dragons.
Nevermind the fact that the Kindly Man hints that the FM may have had something to do with the Doom of Valyria.
Also, I doubt there is some secret, sophisticated way of killing dragons. You either kill them in battle(arrows, ballistas etc.) or poison them, like the Maesters are implied to have done.
It doesn't add up. If there is any lore on how to kill dragons the FM would have had it far before the Maesters did. No, if the theory that Euron paid the FM with a dragon egg is true, it's more likely that the FM are looking for lore on how to
hatch a dragon - since they must know Daenerys did it.
That would be lore the Maesters may have(or at least the FM would think so) since they served the Targaryens at court for 200 years.
edit: ninja'd