Do I need to register my Responsible Person individually in every EU country?

No. It's enough for your Responsible Person to be resident or active in one EU country. From there, they serve as the regulatory interface for all EU markets — at least at GPSR level. However: some member states have additional local requirements, such as registration with national monitoring authorities. You should clarify this with your service provider. The good news: specialised service providers routinely manage these compliance details and know which additional steps are needed in which countries.