I like this idea.Guessing here. It looks like Joe is not using a punch that fits the ID of the bearing and the grease is squishing out around the punch. OR, the punch has a diameter that reduces before it reaches "home" going through the bearing.
Just for giggles, it wouldn't take a lot of time to cobble up something like this to try. Find a bolt that has a head, that when rounded, fits through the ID of the bearing. Grind the points off. Make a saw kerf from the head end into the shank of the bolt. Insert the head end into the bearing so the head is on the back side. Drive a small wedge into the kerf to spread the head so it grabs the back side of the bearing . Take a short piece of pipe, or a couple pieces of steel that will sit outboard of the bearing's OD. Bridge the pipe/steel with another sturdy piece of steel with a hole for the threaded end of the bolt to protrude through. Put a greased-up washer and bolt on it and tighten.
EDIT: Pardon the crude image.