I love the name Rhys and want to use it definitely as a middle name and possibly a first name.
But honestly, on a child if you want the Rh-eyes pronunciation I agree with the above posters about spelling it Rise because otherwise the child and you will be forever correcting the pronunciation for people since the majority of people know it as Ry-ees.
I would even spell a character in a book as Rise because people will read the name as Ry-ees otherwise. Unless a movie is made and people hear the pronunciation you want it to be or on an audio book you tell them how you want it pronounced when they record it. Even then might be a bit confusing for people reading along, seeing it and wanting to say Rh-ees but hearing Ry-eyes said instead and if people are just listening and not reading along also they might hear Rh-eyes and assume it is spelled Rise.
So, sorry imho I would stick with the original pronunciation whether for a child or a book character to avoid any confusion for people.