Despair timeline: Let's say Izuru decided to create a cloning device and tested it on himself thus creating little Hajime. Izuru can't tell why he's so diffirent from him cause he expected more of a resamblence, but he still takes care af the child. He could either call him Izuru as well as Hajime (Let's say that name felt familliar to him and it seemed to fit him). Izuru's fascinated with little Hajime and makes sure that he can develop and grow healthily even if recources are scarce.
At first he would be quite intrigued as for why his clone is so different from him, but after a while he would probably conclude that this is his past life as the child is of course an younger version of himself where he as Izuru wasn’t even existing
I would go with Hajime, either because he remembered that Chiaki had called him that before dying or because as you mentioned it sounded familiar to him and he found it fitting
He would take good care of him, that I’m sure off
Also this AU reminds me of a bunch of artwork I have saved on my computer where Izuru is in the middle of the tragedy and is taking care of a child Hajime. I would link them but the original artist deleted their Pixiv account as it seems...