I have tried to do the same thing , and my experience was not good. Those arbours seem to be really in there tight.
From the few attempts I have made over the yrs, I would just pick up another chuck for the other machine. Pressing and whacking with a hammer cant do it any good.
I had one 3/4 chuck that was being stubborn, and I didn't want to hurt it removing the arbour. I ended up machining off the arbour up close to the back of the chuck. With a hole drilled into what was left , it was weakend enough to push it out with very little force.
For me, I think if you really don,t have to, then don,t do it. The out come could be a trashed chuck. That is just my opinion though.