We tattoo when we brand, usually around 8 months. Unless we are selling before (like pairs), then we tattoo before they leave.
The tattoo fades, so doing it at birth would not work for us. We show at national shows, and the tattoo needs to be legible when they inspect at check in. Many times we have had to redo a tattoo because it faded.
You can register and not tattoo, we do it all the time. You report the number to the registry, and they send papers. You just better make sure the animal is tattooed before leaving your place.