Think there 2 small cut 180 degrees apart out where you'd hit with a punch to knock out the cup. If you could get hook in the cut out and slide hammer it out side to side.
I never heard of anyone removing the race without removing the center section and driving it out from the inside. They don't come out that easily.
It's quite involved setting up the pinion preload. The proper preload on pinion bearings are measured in resistance using a inch pound torque wrench with the center section/carrier removed. The use of shims sets the preload. Removing the yolk to replace the seal and reinstalling it is mostly guess work for the proper preload. Marking the nut before removing it might get you in the ball park when re-installing it but it's just a guess. Also if your rear used a crush sleeve you should replace it with a new one to do it right. Again this should be preformed with the center section removed.
Can you get by with what you're doing? Many have but it's not totally advised. If you plan on leaving that race/cup in there, inspect it for a pit that matches the one on the bearing. More than likely that rear sat for some time and moisture in the top half cause the rust. The teeth on the ring gear that wasn't submerged in lube probably has rust pitting also. If the cup is pitted a new bearing may not last that long either.
Just throwing some thoughts out there.