This idea might end up being really short or really long, but we'll see where it goes.
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I want to see Light, right after his death in the anime, to be transported to the world of Re: Zero and use it as a second chance at life. I want him to have reflected on his past actions, and decide to repent to a degree. This does not mean that Light will be adverse to killing anyone, however. I'll explain more on that later.
I want Light to still have the death note, but for Ryuk or any other shinigami to not be with him. In other words, I want him to still have the death note, and be able to kill people with it, but other than that, I want most of the other rules to be null and void. As for the whole nothingness after death thing, I want that to be interpreted in a unique way, with it meaning that he is trapped in an unknown world, forever destined to die over and over again to help others. (Return by Death).
Light should start off confused, with his mind heavily messed up, when he's transported to the world of Re: Zero. After several moments of just standing still in confusion and trying to sort out his thoughts, he should try and gather information from people. When he comes across the thugs, his first instinct is to use the death note, but before he can do so, he's interrupted by Emilia. Light still thinks he should kill the thugs with the death note, but he refrains, since he doesn't want to get too used to using the death note again.
Light decides to help Emilia, because the part in him that wants to be justice still exists. Him and Emilia search for Emilia's insignia, and when they eventually make it to the loot house. There Light stumbles across Elsa, and he tries to write her name in his notebook, but he doesn't manage in time. In death, he thinks about all of his doings as Kira, realizing how insane he had become.
Following his "Return by Death", Light immediately realizes his new capabilities, though he doesn't yet realize it's a punishment from the death note. He decides then that he'll become justice in a different way this time, only using the death note when absolutely necessary. He'll use Return by Death to discern the enemy's motives, and then he'll judge whether or not they deserve the death note or not.
So that nobody fears Light's power, he decides to keep the death note and Return by Death hidden from everyone. (He doesn't know that there is already a measure in place in case he were to talk about Return by Death, since that's learnt by Subaru in arc 2 of canon.) To help do this, he decides to write elaborate scenarios to kill his enemies, using information he's picked up along his multiple deaths, rather than just using a heart attack.
Light dies again without learning Zelda's full name, but he does not die a third death to the thugs, and does not call Emilia by the name of Satella, because he's already picked up in the first loop that Satella isn't her real name.
This loop where originally Subaru died in canon is when the final loop of arc 1 occurs this time around, and most of everything goes like canon except for a few things. Light decides that he needs to die more times to make Elsa's death more believable, (learning information), despite the fact that he has learned of her name via her telling it to Reinhard. Once Reinhard swings his sword at her, however, he seizes the chance, and he writes in the death note that she dies to Reinhard. The death note does not work on Elsa's curse doll ability though, and so she survives to return in arc 4. This will not work for keeping the plot similar to canon later on, as not everyone has Elsa's ability, but it doesn't have to, due to the nature of the enemies faced later on.
You will have to make up a new work-around for the death note's abilities for the next couple arcs to work. I'd recommend making it so that it can't kill sin archbishops, witches, or the mabeasts. All of these exceptions should be due to the same reasoning; they're either not human entities, or no longer considered human due to the witch factors/ being a witch. The death note should only effect those it considers human, so I think this reasoning will suffice. (I know that's only an assumption of canon, but the death note wouldn't work on shinigami, so I think it's a reasonable conclusion to come to.)
Remember, you don't have to follow the canon story of Re: zero word for word. That is not what I'm saying; I'm merely stating what to do if you want to follow a similar path to canon. You can have it be a completely different idea, with Light joining Anastasia's camp and reinventing things, if you want, or something of that sort. Either way, I'm not going to elaborate on the future arcs as that would make this longer than necessary, and take away from some of the creative freedom that comes with writing things like these. So that will be all.