I want Archer to be transported to the world of Re: Zero. As for how this happens, do not make it so that he's sent to the world on counter guardian duty. Instead, I want him to be summoned by Emilia just as she's about to die to Elsa in the loot house. For this to work, you'd have to bend canon a bit so that Emilia arrives before Felt and Rom are killed, since I don't want Archer to have Return by Death and otherwise they'd be already dead by the time he arrives to the scene. If it wasn't already clear, I want him to replace Subaru Natsuki.
I want him to have been freed from his contract with Alaya because the world isn't under the influence of Alaya and Gaia. (It's instead under Od Laguna.) Do not reduce Archer's strength as a servant to match his opponents; if anything, his power should be increased due to the lack of Gaia's influence on his traced weapons.
I want Archer to have summoned as his true self from the throne of heroes because of him being moved to, presumably, another universe/multiverse entirely than the Nasuverse. This also means that he shouldn't be limited to just the Archer class, and that you are free to show off other potential abilities he may have that weren't shown in Fate/stay night. He should still be, technically at least, the servant to Emilia, despite all of this, as it gives him a reason to stick around plot-wise.
I want him to have the level of power Alaya gives him for counter guardian duties, even with the contract having been severed. Basically, Archer should be second only to Reinhard in terms of power scaling in the world of Re: Zero. He has to be powerful enough to be able to get through the plot of the story without dying or getting any upgrades to his abilities, (other than the the restrictions imposed by the Archer class and Gaia being removed), even once, after all. If you don't like that, you can have him trace some of the weapons he sees from other characters along the journey. Other than that, you could also have him weaponise authorities or have him learn some of the magic in Re: Zero.
I want Archer to fight Elsa and win quite easily, though I also want Reinhard to appear at some point so that he gets to meet Felt and find out that she's a royal candidate for the throne. I don't want Elsa to die here; Archer has no prior knowledge of her regeneration ability, so she should be able to escape in time. She's also important for the plot of arc 4 of the story, so she can't just die there like that. At some
point or another, Archer also should explain what his position in regards to Emilia is, (her servant), and how his powers operate, minus an explanation of Unlimited Blade Works.
Following all that, Archer comes along with Emilia back to the Roswaal mansion, since he's her servant and all, and then the following day is when arc 2 begins. This arc may just be the arc that differs the most from canon, for reasons I will now explain. First: there is no witch's scent, so Rem has no problem with Archer. This takes care of what is undoubtedly one of the biggest issues that occurs in this arc. Second: Archer can dispel curses, and he can easily tell whether or not someone has one or not. This means that the moment he notices the mabeast dog can curse others, he will take care of it then and there. This arc should fly by very quickly for these reasons, and it should be a very minor conflict with no real development unlike how it was in canon. This should majorly affect the plot later down the line.
In arc 3, Archer accompanies Emilia to the gathering of the royal candidates, and similar
to Subaru, but with harsher words and more tact, he should speak up against the discrimination happening to Emilia. Unlike with Subaru, however, when Julius fights Archer, Archer wins, and from that point on he is considered Emilia's unofficial knight. This also changes many of the events that would otherwise occur in arc 3 of the story.
The natural progression of the story from here would be that Archer would not go to Crusch's place, since he wouldn't have to split up with Emilia. Instead, for plot convience, have him decide to visit their camp for political reasons, and have him decide to return after a couple of days. He stays there awhile, and then he returns as he initially planned. When he arrives, however, the Emilia camp is under attack by Petelguese and the witch's cult.
(Since I can think of no way for this to go in a natural/realistic manner, this is what I'm going to give you to base your story off of, but you can change it as you deem fit. The problem that appears here will be solved in a moment so that it doesn't occur in arc 4.) I want Archer to Ty his best to drive off of the witch cultists, but without prior knowledge of the attack and a sufficient army behind him, he has arrived too late to be of any real help to the situation.
He does not die, but nearly everyone else does. Just as all seems lost, however, Zelretch makes a cameo, bringing along with him Aoko Aozaki, the wielder of the fifth true magic, time magic. She turns back time for Archer so that he can fix the situation, and Zelretch gives Archer a much lower-level version of Gilgamesh's clairvoyance that only activates when something bad is happening to people he is acquainted with. (This is so that he won't need yet another work-around for Return by Death later on in the story.)
Archer tries to disguise his newfound clairvoyance ability as a divine protection, and he goes to Crusch's camp to acquire their help in defeating the witch's cult. (How you go about Archer acquiring the Anastasia camp's help is up to you, since I don't know how to implement that.) After trying to get crusch to help out of basic human decency and failing, as well as Crusch becoming suspicious due to seeing through his lie about his divine protection, Archer says he will help the camp exterminate the White Whale in exchange. (He has learned of this during the time he was at their mansion, so make sure to include him finding this out during that time frame.)
Archer states that the clairvoyance he has is just a type of magic, and that he just wanted to see if she'd call him out on his lie or not. This is indeed what he plans to test out her ability to differentiate lies from the truth, as he has already been told of her ability when he was staying at the mansion and wishes to test it out. Thus there is no lie, and a temporary alliance is struck between the three camps of Emilia, Crusch, and Anastasia, just as in canon.
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I'd write more, but this is already too long as it is, so this will be all.