It's sort of a given that here on Bike Forums Bicycle Mechanics posters will suggest and claim they use
anything but the factory lubrication. Shrug. Anyway: Shimano Internal Hub Grease Y-041 20600
Google is finding some YouTube videos on the disassembly/reassembly of the 3CC.
I would agree that factory grease is good. But that requires disassembly, and 50 years ago, that involved an incredibly large Proto adjustable wrench (like for bolting together skyscapers), like 2 or 2-1/2" wrench flats. Although if you have a bench vise, that can also work. Maybe on Shimano, not needed, can't remember, I had both Shimano 3 and Steyr 3 (S-A copy I think). For most home folks, taking apart an IGH is above their pay grade. Which is why I think dripping in a good quality semi-liquid lube through the shift hole is a great idea.