In one of my BE supertypes, I have an abstract property ala val inv: FixedItemInv. To save the NBT in my toTag method I have to check if it's a specific subtype or few (if (inv is DirectFixedItemInv) inv.toTag(tag) else if (inv is FullFixedItemInv) inv.toTag(tag) // etc) which is pretty unweidly (also non-future proof). It would be nice if it was somehow moved into a super interface or a higher up class.

That sounds reasonable - something like:

public interface Saveable {
    default CompoundTag toTag() {
        return toTag(new CompoundTag());
    }
    CompoundTag toTag(CompoundTag tag);
    void fromTag(CompoundTag tag);
}

and then make all LBA inventory implementations (DirectFixedItemInv, FullFixedItemInv, SimpleFixedFluidInv) implement that?

Yes, that would very ideal.