Actually, from a purely rational perspective, the first choice is definitely the best.
Because "Option 2" will offend Furuta Nimura, and "Option 3" is even more unwise.
Amou Byakuya himself has already fused with Uchiha Madara and obtained abilities. Being experimentally modified into an artificial half-ghoul is completely unnecessary.
In the early stages of the original story, Kanou Akihiro used Kamishiro Rize as a donor for over a thousand experiments, and only Kaneki Ken, Anku Kurona, and Anku Nashiro—three test subjects—truly succeeded. They were reborn as "artificial one-eyed ghouls." The experiment's success rate was only 0.25%.
No matter what, Amou Byakuya wouldn't take this risk just to become an artificial half-ghoul and get 50 Fusion Points. If the experiment failed, he would either die or be crippled, and the failure rate was astonishingly high.
As for the problem with "Option 2," the reason it's said it would offend Furuta Nimura is that this "steel beam drop incident" was actually orchestrated by Furuta Nimura himself.
In the original story, Kaneki Ken "stole" Furuta Nimura's "first love" (Kamishiro Rize), which led to Furuta Nimura plotting this incident out of jealousy.
It seems to be due to Furuta Nimura's possessive and petty personality, yet Amou Byakuya believes this was just Furuta's excuse.
Analyzing it from a deeper perspective, one can discover that Furuta Nimura had long been filled with pessimism and despair for this world. Even if Kamishiro Rize hadn't done those things, everything Furuta Nimura planned later would still have happened.
He even intentionally pushed Kaneki Ken to ascend the throne in the later stages of the original story. On the surface, it seemed like he was opposing Kaneki Ken until the very end, but some details reveal clues: from the very beginning, he had prepared for two possible outcomes.
One was his own plan succeeding, turning all humans in Tokyo into Ghouls by using the "Dragon," thereby thoroughly resolving the conflict between humans and Ghouls (as everyone would be a Ghoul).
The second was being defeated by Kaneki Ken, with the "Dragon" also being killed by Kaneki Ken. Kaneki Ken, becoming the One-Eyed King and the savior, would possess extremely high prestige, enabling a temporary, relative peace between humans and Ghouls.
However, Furuta Nimura himself probably preferred to achieve the outcome of destroying the world. After all, turning all humans into Ghouls also thoroughly solved the conflict, and there were actually solutions to the food problem.
Of course, this was all Amou Byakuya's speculation, with no particularly reliable evidence.
In any case, Furuta Nimura had to create the "Dragon," otherwise, neither outcome could be achieved.
To create the "Dragon," a "Kakuhou" was needed as the core. The only candidate for the experimental donor was Kamishiro Rize, the "Washuu" with the purest bloodline, because only the Washuu bloodline could "become the Dragon." Additionally, a half-ghoul needed to be artificially created, and this half-ghoul needed to have Kamishiro Rize's Washuu bloodline.
Historically, a natural-born half-ghoul appeared in the Washuu Clan, known as the true "Original One-Eyed King." Only he had successfully "become the Dragon." Everything Furuta Nimura did was to imitate the "Original One-Eyed King's" process of becoming the Dragon, so Rize became a necessary sacrifice.
As for Furuta Nimura's personality, it's hard to say if he was really that petty.
If he truly had a strong desire to control Kamishiro Rize and was very petty, Kaneki Ken would have been killed during that "steel beam drop incident." It would have been incredibly easy for Furuta Nimura to take revenge on this "love rival" who had no background.
In summary, if Amou Byakuya saves Kamishiro Rize now, it is equivalent to obstructing Furuta Nimura's "Dragon" plan, and it will inevitably offend Furuta Nimura.
"As I thought, I still can't just leave her..."
After careful consideration, Amou Byakuya still decided to go back there and save Kamishiro Rize.
He didn't deny his double standard. If it were any other Ghoul he didn't know who wanted to eat him or even cruelly toy with him, he wouldn't just not save them; he would definitely kill them if he had the ability.
But this was Kamishiro Rize, a character he found tragic and pitiful.
After becoming an experimental donor, Kamishiro Rize's fate was extremely miserable. Her life was maintained with a nutrient solution that could barely provide sustenance, because a Ghoul's regeneration ability is so strong they don't die easily.
Undergoing countless experiments, her "Kakuhou" organs were extracted innumerable times. The physical torture, combined with the long-term lack of eating human flesh, caused Kamishiro Rize to later become delirious and insane.
In the end, she became the core of the "Dragon" and was only freed when she was killed by Kaneki Ken.
When Amou Byakuya saw Kamishiro Rize's ending, he personally felt it was a great source of dissatisfaction.
Now that he had crossed over into this world, how could he not change the regrets and dissatisfaction in his heart?
Arriving at the scene of the incident, Amou Byakuya's gaze was fixed on the area where the steel beams had fallen.
Looking at the tragic scene, Kamishiro Rize was pinned beneath several steel beams, having already fainted.
He used his full strength to move the steel beams off Kamishiro Rize one by one. Fortunately, he possessed Madara's taijutsu, which also gave him physical fitness far superior to that of an ordinary person. Otherwise, steel beams of this weight were simply not something an ordinary person could move.
With nowhere else good to go, Amou Byakuya took Kamishiro Rize directly back to his home.
[Ding! Host has completed Option 2, received reward: 10 Fusion Points!]
He placed Kamishiro Rize on the bed to lie down, while he sat beside her, slowly waiting for Kamishiro Rize to heal and wake up on her own.
The first moment Kamishiro Rize opened her eyes and saw Amou Byakuya, she instantly remembered being knocked out by the steel beams. She subconsciously thought Amou Byakuya had ambushed her.
Therefore, she abruptly stood up, her Kakugan emerging, and four Rinkaku Kagune also instantly burst out from her waist.
Seeing Kamishiro Rize awake, Amou Byakuya reminded her in a calm tone, "Calm down first. I was the one who saved you."