Your Jounin may aptly return when in need if no one else wants to, and I will offer you priority over anyone else asking to make a Jounin, but I do want to keep a 1:1 ratio for now.

Your Hyuuga presently should not even know his nature release. During the Reign of Sai, he was most likely in hiding and training, which is why he is a rather old genin. Chances are, he probably didn't even know most basic ninjutsu; just trained with the Byakugan and Gentle Fist. The Hyuuga, due to their doujutsu, were hunted down extensively, as were the Uchiha. There also is no shame in not having a full roster of jutsu. I put in six as a limit because I know there are those who would push it, but reality is 3-4 is a decent amount. Look at canon:

Shikamaru essentially only knew Shadow Neck Bind; Chouji only had the Human Boulder and Size manipuation; Ino just took over minds. Effectively, they each had one or two hiden, then just academy jutsu. Neji, a prodigy, only knew the Byakugan based abilities. Hinata knew the same ones, just significantly weaker. 

Just focus on your Hyuuga being a Hyuuga.Make him a strong character for the setting, not a strong character for an arena.