I don't know much about Black clover, so really I'm fine with you starting anywhere in canon with that series, though the beginning of canon would probably be preferable. Basically, my idea is that Archer somehow gets free from his contract with Alaya, whether it be via Zelretch or something else. He gets reincarnated into
the world of Black Clover, but his parents end up dying in a fire similar to the one the person known as 'Emiya Shirou' was created in. Now, from here I have multiple paths outlined in regards to what could occur. Option one is that Archer gets taken in by the orphanage and grows up alongside Asta and Yuno as a rival. Option two is that he grows up as the prince of the Clover Kingdom, and option three is that rather than getting taken in by the orphanage he gets saved by the Wizard King, and gets trained to work as his assistant/advisor, or whatever works as a position next to him, when he gets older. I think this idea has a lot of potential, though I've said that nearly every time so far, and from any of those options, you can go in whatever direction you'd like.
Honestly, although I keep listing multiple scenarios and things of that sort in these ideas, I do actually encourage you to not follow those ideas too strictly, as I am rather a fan of the unpredictability of where these ideas can lead. These ideas are just supposed to be a starting point, if anything, to just branch off of, and work from there.