This topic is dangerously close to story-ideas, I think.
For speculation's worth: In the Dark Mirror
permutation of the MU, the United Empire of Planets is so powerful it probably would've crushed the Furies with ease.
Basically, we don't know when the MU diverged from the Prime Universe - and as such whether the Fury society ever formed or survived.
However, speculating is rather pointless, because you could probably ask for virtually every situation "How did the MU people deal with that?" - Mirror V'Ger, mirror Khan, mirror Porthos incident on Kretassa? There are works that tackle some issues exemplary, e.g. Drema IV's survival, Shattered Universe
with several TOS threats, Ligon II
in Dark Mirror, the Trill Parasite Crisis etc.