Rey needs to be referred to as a Palpatine or the whole point of my original idea was for nothing. And of course, in order to do that he has to state she is his heir.
I’m also unsure about Kylo knowing about Palpatine’s desire to possess one of them at this point in the movie. Especially if we were to remove the “Kill me, and my spirit will pass into you” stuff and put it back where it belongs. This moment would then just be more about who Palpatine will see as his rightful heir when they fight. Then the dark twist that he just wanted to possess the “winner” would come later.
As a result I’d say what I have for my original last two paragraphs is what I’d prefer. It’s also not very clear now that the place her parents are looking for is on Jakku, I was under the impression we were trying to explain Lor San Tekka’s village? Also, Rey’s reaction of “Don’t” makes less sense to the generally inoffensive statement you have there.