Thurston or Barron. Nickname could be Tray, as he's the III.
When it comes to wealthy people I think of retired old people with millions
J.R- Dallas, He's like J.R Ewing II by the way
i think names ending with son or ton would fit in well with his social status..
good luck in finding the perfect fit
Reginald or Regis (Reg or Reggie)
Bennington (Ben, Benny)
Baron (Barry, Ron, Ronnie, Bear)
I also like the idea of Tripp since he's the third. Two middles could be fun.