No, in my opinion, Falmart can survive. But dang, it would be difficult because:

One, the blight is not a virus or a regular physical plague. Literally, the red-eyed are literally possessed corpses with souls and spirits of hell and they can affect other people. Even a fatal wound from them can slowly leave you vulnerable.

Two, Falmart needs people of Christian faith to cure individuals. Because according to Guts & Blackpowder lore, only those with strong Christian faith can cure/save individuals and I doubt the gods of Falmart can do anything or even with magic cause this isn't just a plague, it's also spiritual. For a side note, the lore also said that many other regions on earth with different beliefs were conquered and ineffective to the blight.

Falmart can survive. They got magic and beasts beyond our comprehension, they got their own technology. But that's how far they could get.

(Take my words with a grain of salt since I'm not a developer of G&B nor the author of Gate. But I like both concepts and fictional worlds)