If he loses his right arm, and Doppelgangers have to look exactly like him, they also have to lose their right arms

If Putin loses an arm, all of his doppelgängers will need to lose an arm too. Otherwise they’ll be immediately identified as doppelgängers.

It’s a pretty well known thing that Putin frequently uses lookalikes for public appearances. It’s apparently an old habit from when he was a KGB agent. So if someone snipes him from a rooftop, it’s probably not even him; He can claim to have miraculously survived the shooting, then carry on as if nothing happened.