As Will says you need a long throw air hammer.
He mentions a 3x hammer, but there are two types of air hammers. The X series are aircraft standard used to properly swege rivets. There are special rules and you have to use the correct hammer for the size rivet.
Then there are the 'automotive' kind.
The difference is in the diameter of the working hammer. The automotive hammers use a larger diameter which means more pressure per square inch and probably more mass so you get more energy per blow.
The X's relate to the length of the barrel, the 3x is like 3" long (been a while and that may not be perfectly correct). Automotive is like that too. The cheap ones are close to 1" long.
I happened to pick up a 4" throw automotive air hammer at a garage sale insanely cheap. I needed to get some perches out. So I cut off the end of a chisel tool and put the axle in my 150 lb vice (big vices are a wonderful thing). My brother who is heavier than I put his weight into it and pulled the trigger. It did not move right off, then it slowly started to move. The first one we tried has been cut off before I got it. So with our surprising success we moved to the good perch. Put a nut at the end to help guide the punch end. Same thing as the first perch. We had to take the nut off to push it through, but it cooperated.
The end of the perch was visibly mushroomed, but the nut was able to still be screwed on by hand!
So the right tools seem to the do the job.
The key ingredients from out success.
4" automotive air hammer (or one of the other long throw hammers). These can be expensive to buy. Learn what they look like and hope to find one cheap at a garage sale or flea market. You should be able to find a mechanic that will help.
Firmly mounted with zero movement axle. Do not discount firm mounting. A very small amount of movement could prevent the perch from coming out.
Solid well back holding of the air hammer.
Final rules of perch removal.
DO NOT heat the axle any more than the point where grease starts to smoke. The axles are heat treated and loose a significant amount of strength when heated.
Do not expect penetrating oils to do any good. The rust is a solid mass and they will not penetrate unless the perch is already loose enough to come out easy. Heat may allow the oil to penetrate, but see rule above.
A final note- it was a pain in the rear locating the information about air hammers when I looked a few years ago. The aircraft ones are easy to learn about. The automotive ones I learned about by finding the spec sheets from Chicago Pneumatics (I think).
Hope that was not too much information.