Yes, it would be asinine on his part. And yet, rushing headfirst into situations that he can't handle without first thinking it through is exactly his M.O. See: his blackmailing attempt. It is absolutely in keeping with what we've seen from him before that he didn't plan this out thoroughly. Before he goes to TV, the last thing that happens with him is he walks away in defeat after giving up the Tumbler plans. He knows he lost. You think Fox kept him employed there after the blackmail attempt? He later decides to change his mind and go on TV to reveal Bruce's secret. Sure, if only he was the type of guy who thought things through, I might agree with you that he planned ahead to have copies of the proof, even though he gave it to Fox. But it's established that he isn't that smart. I guess he "evolved" into a smarter guy (retroactively, of course, since he would have had to have made the copies BEFORE he gave the plans to Fox---which means he would have learned his lesson that he should plan better from the confrontation with Fox---
before his confrontation with Fox
)
It's interesting that you are perfectly willing to accept something like this probably happened offscreen
(even though we see no aftermath of this being the case onscreen), yet you refuse to accept that the LOS discovered the existence and precise location of Batman's armory offscreen
(even though we DO see the direct aftermath of this occuring onscreen). Your argument that Reese has copies hinges on but in the second instance, my argument that we don't need to see exactly how the LOS discovered the location of the armory is