I am not familiar with the rules for names like this. If you don't mind me asking, how does it work? Are your first male descendants expected to be named the same until one of them changes his mind?
I'd have to check to be certain, but I am pretty sure we didn't put it on birth certificates. It's convention only - some businesses (notably banks) have a "suffix" field to help disambiguate customers, but many of them are drop-downs which stop at "IV" for some reason.
What really makes me feel like royalty is not the name, it's the throne room I had installed in my summer home in the Bahamas. That and requiring everyone I meet to call me "sire." (/s)