What you are asking, I think is not advisable. To remove the tile without damaging the mud base that is over the hot mopped pan is really asking for trouble. To crack it only in just one spot, will require a repair at least, but it would be better to replace the whole pan. But that itself is its own issue. Removing the whole pan would surely pop a hole in the pan. Then you will be into removing the wall tile trying to get to a good place for a waterproofing tie in. That in itself again is tedious at best and doing that while not chipping the neighboring tile??
To me, the only possible solution would be to remove the drain cover, roughen up the existing tile, (right angle grinder and a lot of dust comes to mind), set an extension on the existing drain, lay new tile and reset the existing drain or a new one like the old one that will fit and look new.
To replace the drain with a totally new one will not be possible, since the hot mopped pan is tied to the existing drain. To mess with that to replace it would be disastrous.
Sorry for the doom and gloom, but shower pans and tile are usually, when done right, are a once and done type of thing.