Also sorry if I am stating the obvious but the piece of pivot steel you would be putting into the hole needs to be of a slightly larger diameter than the original pivot as by the time it is polished and burnished it will take the diameter down a fraction. If you have enough meat on the arbor you could even get away with inserting a much bigger piece of pivot steel and then take it down to near size with a graver leaving enough to polish/burnish to size.
This is assuming of course you have a watchmakers lathe, you will find this very difficult with a dremmel as it spins way to fast. If you get stuck, send it to me and I will do it for you.
why on earth do you think it impossible to put a hole dead centre in an arbor? I am no machinery expert but I have done quite a few re-pivoting jobs and find it quite simple? What method have you tried using that has caused you a problem?