I don't know what you mean when you say "the Live center has a #3 Morse taper" Does the tailstock have a #3MT unlike most of the other wood lathes? Both of my Oneway live centers have #2 Morse tapers as do the headstocks on all of my wood lathes.
I made some very accurate cup centers and other adapters to screw onto the Oneway live center from both wood and plastic. I first cut a blank larger than I needed and drilled and tapped for the 3/4" thread. Then I mounted the live center in the headstock and prevented it from spinning with a piece of iron wire through the locking hole. I screwed the block onto the thread and turned it to suit. Threading first and shaping second forces t
he wooden and plastic centers to be perfectly concentric. I've made cup centers, special jam centers, and some with pins to turn things with holes drilled in one end. Here is a picture of a couple where I turned pins to fit some project or another.
Attachment 419563
BTW, I far prefer the Nova live center when I make special adapters. It's the most versatile live center available in my opinion. This is what's in the box:
Attachment 419565
The Nova center has a short #2MT socket which avoids the need to drill and tap. I simply turn a short MT on a piece of wood, mount that directly in the headstock, and turn whatever adapter i need. Here are a few wooden inserts along with some that come with the live center:
Attachment 419564
JKJ