I don’t claim to know anything about how this works in WoW lore, but “killing” a personality would mean that the Lich King would lose the Ner’zhul part and all that entails, leaving it as just death knight Arthas with the helm.
As we still fought and killed the “Lich King”, and he still seemed to refuse to identify as Arthas, it would seem the personality was only suppressed for as long as Arthas’ body and mind remained living. Otherwise, Bolvar would not have fused with anything.