The main thing debunking this is secret report #12, which clearly says that Luxu gave away his keyblade before doing any body-hopping, but in the face reveal scene he still has it. Another is secret report #11, where the talk about the virus and "the same trick" has no explanation in this theory, but starts making sense if you assume that Luxu himself is the virus.
So, I'm going to spring for an unorthodox reading of that report. The exact line is: "Amid the chaos, I bequeathed my Keyblade to one of the Union leaders, just as the Master instructed." To me, it's not impossible that Luxu was being cheeky here and fudging the truth. He technically did give his Keyblade to Brain, one of the Union leaders—he just also took his body in the process. What he's saying is technically true, but it could be leaving out a lot of details. Luxu/Xigbar feels like the kind of person who would do stuff like that.
And then, of course, we also have to consider the ever-likely explanation that Nomura just didn't know what he wanted to write when he made those reports. Nomura has exhibited time and again that he doesn't think of everything, even though he always tries to make people believe he does. It's quite possible that the idea of Luxu taking over Brain's body and sending his heart into the future was a later idea (keep in mind how much time there was between the release of KH3 and the ending of UX, also the ending was stalled because of extra work that had to be done on it, i.e., maybe story revisions).
My guess is, Nomura thought of a new story he could tell with Brain in the future (or maybe Square just wanted him to make another mobile game), and in order to tell it, he had to retcon (as he so often does) a piece of lore he had established earlier.