It seems the scripts don't allow the cloak to appear right at the beginning of the third mission; I think this would be the case even in the original game (if you were to give it to yourself using cheats). The Necromancer campaign is super easy even on Hero difficulty and without such exploits. Do you think it's worth fixing this in the mod?
Generally, yes, it probably should be.
I did everything as described above for the Necromancer campaign, and it played out perfectly. In the Elf campaign, everything is fine with the artifacts; there are no issues with the Dwarves up to the Greater Vampire.