This is related to https://kagifeedback.org/d/1034-share-this-result-page-probably-ignores-website-ranking-adjustments.
One of the biggest differentials of Kagi is the customization. The idea is to create an extra option to allow users to share the result page exactly as we see, including custom ranking adjustments, URL rewrites, custom CSS and appearance options.
Of course, this should come with a trivial UI (nobody likes an extra button for everything). I'm not sure how to do it, but I'm pretty confident there is an elegant solution.
An additional point is the privacy, even though there shouldn't be a lot of private data in someone's search engine customization, it is for the best if the default share behavior stays as it is today while a "full share" needs some extra click (so nobody clicks on it unwillingly).