the Byakugan
may not be as efficient as the Sharingan in terms of anticipating movements but it should be enough to counter it. let's say that a Byakugan user and a Sharingan user (with the same states in physical and mental abilities ) fought. if the Sharingan user can anticipate the Byakugan user can anticipate the Sharingan's user counter attacks which allow him to react to it. the Sharingan user also would not be able to do
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, because that would be anticipated and perceived. Naruto didn't have the ability to anticipate movements, that's why he was at a disadvantage and the fact that Sasuke was also superior to him in terms of skill.
also, the Byakugan is superior to the Sharingan in terms of insight so it should be able to perceive movements better. so I doubt that the Sharingan precog would grant a significant advantage against a Byakugan user.
however, if the two went against other in a raw Taijutsu fight with only the assistance of the Doujutsus then the Sharingan user would win comfortably, but if you gave the Byakugan user the Gentle Fist then he would obviously beat the Sharingan user with ease.