I think Byakugan would win too, Sharingan can copy jutsu's and physical movements of taijutsu too as well as cast really strong genjutsu's but the Byakugan can see the chakra in the person so genjutsu doesn't work on them since they can just activate byakugan and see the chakra of the person casting the genjutsu right through the genjutsu and I don't think the byakugan style of taijutsu can be copied so thats another mute point for the Sharingan.
Unless the Sharingan user has copied tons of awesome jutsus and stuff I don't see them ever beating a Byakugan user based solely on their Kekkai Genkai alone. The Byakugan can also see 360 degrees and if at the Kage level Im sure they can see the Tenketsu's too, and once that Sharingan users Tenketsu's are sealed up hes done for, no chakra for jutsus's, a weaker taijutsu form, and genjutsu that doesn't affect the byakugan user.
I also don't think weve even seen the fullest extent of the Byakugan, im sure they have a higher dojutsu level too and probably secret jutsus too like the 3 secret jutsus of the Mangekyo Sharingan.
Dunno if Tsukiyomi works on a Byakugan user since its like their transported somewhere else in their own minds but Im gonna say it probably doesn't or at least can be broken out of by a Byakugan user.
I vote Byakugan