He already has a byakugan in the future.
You're right, awakening the tenseigan requires "undiluted" otsutsuki chakra and a pure byakugan. The tenseigan vessel or whatever is it's own separate thing.Wasn't Tenseigan achieved by combining the two blood lines descended from Hamura? The same as achieving Rinnegan...except the body side stole from the eyes this time.
The thing that took thousands of eyes was their doomsday weapon.