Hi
Do not just hammer it out. Drill the riveted end with a drill slightly larger than the axle dia until you come down to the bracket and a tenth of a millimeter into the bracket.
The axle will then come out using very moderate force and not distort the bracket.
Go to a well stocked nuts/bolts shop and scrounge around for rivets until you find with same dia and long enough. Saw it off two millimeter longer than what the bracket is wide and after all is restored insert and rivet it in place.
If you have a lathe , center drill it like done on the original, makes it much easier to rivet.
Ola