The last time I did an idea of this sort, it was one that had Bleach in the mix as well, so I decided this time that it'd just be an Archer x RWBY idea.
I want Archer to be reborn in the world of Remnant. How he gets there can either be via him being summoned there by the counter force, or via Rin somehow transporting him there to save him from his contract with Alaya. Another possible option, as he is always an option, would be Zelretch transporting him there.
Anyway, the idea is that Archer is ridiculously powerful in comparison to the characters in RWBY, and that he wreaks havoc on Salem's plans on a whim. You could have it so that it's just due to a power scaljng difference between the worlds, but I have some other justifications if you want more. 1: if Archer is empowered by the counter force, then that's the only justification you need.
2: Gaia might be gone in the world of Remnant, if you want to go that way with the story. If so, then Archer's traced weapons would last indefinitely and the strength of them would also be significantly increased.
I want him to either grow up with the head of the Schnee family, Summer, the head of the Arc family, or Ozpin as his guardian/parental figure. (I apologize for not knowing some of the names of the characters in charge of the families.) The story can go many different directions depending on which of these people you choose.
The Schnee family would make the most sense given Archer's white hair, but if you want to go the direction of him looking like Shirou again, then he could really fit anywhere else other than the Schnee family. Going with that family could make Archer have an interesting sort of dynamic with Weiss, where she's nicer around him, but on the other hand there are other options that are equally as intriguing.
If you choose Summer, then Archer could have a dynamic with Ruby and Yang where he's closer to them due to Summer's death. Going with the Arc family, you could have him secretly train Jaune, making him actual promising hunter by the time he goes to Beacon. If you go with Ozpin, then Archer gets to know the truth of Remnant earlier, and the story can go directly into the heavier elements once it's around the time of canon, or perhaps even earlier.
Once Archer overhears Ozpin talking to his associates about Salem, or in the case that you chose Ozpin as his guardian just immediately once given the opportunity, he should reveal who he is and where he came from. Not to everyone, though, just Ozpin and his associates, and maybe the others later depending on how the plot goes.
Beacon shouldn't fall in this story, as Archer should take of Cinder before that happens, or perhaps if you still want it to happen, then he should deal with Cinder as it's happening. Either way, Cinder shouldn't last as an enemy past the point of Beacon's fall. I also want Roman Torchwick and Neo to be "convinced" by Archer to work for him instead of Cinder, and for them to eventually become close Allies rather than enemies of their own free volition.