Neji wins this.
Gentle fist can not be healed by ones own self using their own chakra unless the chakra in their hands are not disabled. Byakugou heals from the inside out by spreading healing chakra throughout the body by means of the chakra network. If the chakra network is blocked, it cannot heal damage to the body, and since the majority of nejis attack are internal in nature by attacking the chakra network, its not like there will be any damage to "heal"
Dont get this ability confused with narutos regeneration and unblocking of Narutos chakra network by kuramas chakra.
1) kuramas chakra is a separate chakra that kurama controls
2) kuramas chakra is also capable of being becoming pseudosentient on its own.
As a separate chakra source independent of Naruto, Naruto was ONLY able to continue fighting because Kurama had helped unblock his chakra network with its own chakra.
Neither tsunades or sakuras chakra is even remotely close to being like that of kuramas because it is not controlled by another person nor is it capable of becoming sentient on its own to unblock the chakra for them.
This means that byakugou is USELESS against the gentle fist techniques. Healing may ONLY be done if the persons hands and chakra system connected to to them are unblocked, though Neji is skilled and smart enough to go for a persons arms to disable them of their ninjutsu as he bas often done in the past.