The piece of metal appears to be part of the magazine assembly and is not supposed to go into the chamber. I would call it a "follower" for lack of an actual Browning term. I believe #22 is the firing pin. Not the same shape.
It is usually held by the magazine spring, which may be damaged. Something had to fail (rust?) to permit that follower to fall off.
As these aren't listed as separate parts, I'd try to replace the entire magazine assembly.
(A quick inspection of mine doesn't show any easy way to take any of it loose...)
.30-06 Springfield: 100 yrs + and still going strong