Doomsday has adapted to many things, I think you haven't read the comics yet, but he survived Darkseid's Omega Beam. The laser beam was stated to be capable of vaporizing both the body and the soul OUT OF EXISTENCE.
The Omega Beam isn't just any regular bypassing durability beam at all. It's a beam that erases you and your soul out of existence. The Omega Beam is a soul and conceptual erasing weapon.
This isn't the first time Doomsday survived soul-erasing or soul-stealing items or abilities.
There was that one time in the comics where he fought off someone who wielded a katana blade that could steal away a person's soul, but instead of being able to steal away Doomsday's soul, the blade simply just collides with his skin, and he's still alive. The wielder of the katana blade states that "Damn this monster! If it has a soul, my sword will take it!"
Another instance is where someone name Deadman tried to possess Doomsday, only to end up being rejected by the monster, and the person who tried to possess him states that Doomsday doesn't have a soul.
With the evidence given to us, it's safe to say that Doomsday is neither a being who is alive nor dead.
Yes, it's true that the likes of the Goal of All Life Is Death and World Class Items that can erase your existence would work. However...
The WCI Longinus stated, "It has the power to completely remove any target from existence at the expense of the user's existence. There was no way to restore the data of anyone deleted by the World Item other than by using the resurrection powers of other World Items; neither cash items nor resurrection spells would work. If someone were to use it on an NPC of Nazarick, it would even reduce the maximum creatable levels of NPCs — the special feature of a guild's home base.
I guess it could work, but why would Ainz even risk using it while he knows it can erase his existence out of reality? Or the guardians, he wouldn't risk losing them either, but that doesn't matter even if he forces or asks someone to use it since Doomsday would focus on human and vassal states before aiming at Nazarick.
And the Goal of Life is Death is also useless, considering Doomsday survived a durability negation beam that erases the soul and its body out of existence.
Darkseid stated that those who get hit by his beam would "Whatsoever is touch by my Omega Beams not only ceases to live, it ceases ever to have been! There is no afterlife, no chance of rebirth, or redemption. Only the eternal agony of unbeing!"
Not only that, Doomsday has greater strength, speed, durability, agility, and infinite stamina, which is vastly superior to physical and close-range combat NPCs of Nazarick, including long-range, considering he was also stated to be capable of deflecting attacks towards his enemies.
He was said to be capable of moving at massively faster speeds than light.
Doomsday also managed to develop the ability of high-speed flight to catch up with Superman.
However, looking back at Ainz's hax, I decided to create two different endings.
The first ending is called Nazarick's Victory where Ainz would be lucky enough to use the Longinus to erase him from existence.
The second ending is The End of All But It, where Ainz gets killed by him. Of course, if you know, in the battle between him and Darkseid, Doomsday didn't allow his enemy to have time to re-charge themselves to attack Doomsday. He instantly pummeled them before they even got a chance to make a second attack, which, in the second ending, Ainz has his jaw dislocated before Doomsday breaks his item.
I told you, even if Ainz does have hax, Doomsday's speed is said to be a lot of times faster than light, it once even traveled at faster than light speed through the galaxy and survived planet-destroying attacks. His speed and strength could've given him enough time to stop Ainz before he even uses any attacks.
The third ending is called The Worst Virus Infection Ending, one of the latest versions of comics Doomsday is where he carries a virus that turns anyone who sniffs it into another Doomsday that would soon have the same mindset as he is. This ending is worst because you already know why.
That's how it goes.
Also, another thing here is that I told all of you how many times, if you don't like the fic, don't read it, why bother coming across it and review it if you know major character would be dying?