My family's tradition is that all the men share a middle name, and all the women share a middle name, so there's still name inheritance even for younger siblings! (There's an OOP joke in here somewhere.)
Amusingly, my wife also has the same middle name as the women in my family, so the tradition is likely to continue.
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)
Also, continuing similar tradition, I hope you've reserved mail addresses ending with 6,7 and so on for future generations.