Never heat the axle past the point where oil start smoking you will ruin the axle's metal. This will make it very soft.
I had a pair of perches that would not move.
Keep in mind I have this big vice and tried all the common things. Though kroil I never bother with as it never actually soaks through if they are really stuck. That is another story.
Not long before this I picked up a pile of tools insanely cheap at a garage sale. In this pile was a long throw automotive air hammer. So I thought, lets give it s try.
I cut the end off a chisel tool and put the nut on the end of the perch. Pulled the trigger and much to my surprise after a short bit it moved, slowly. Got it down and pulled the nut off, keep in mind it was still very very tight. Pushed it through. The end was mushroomed a bit, but the original nut still fit on it. The other side worked the same way.
A KEY part of this, I have a vice that is around 100lbs. Two people had to lift it on the bench. This did not allow the axle to move even a small amount. If you can not solidly back the axle while hammering it will not work very well.
Good luck.