I guess it just doesn't serialize it, because I don't even see anything resembling auto-export (and apparently input slot stack size) in blocks' NBT (using /data get block _ _ _)

Probably also happens when chunk gets unloaded (haven't tested, my base is chunk-loaded)

Versions:
MC: 1.20.1
Fabric: 0.16.10
AE2Things: 1.3.2

Looking at src/main/java/io/github/projectet/ae2things/block/entity/BEAdvancedInscriber.java it appears that you just forgot to serialize/deserialize those properties