The byakugan is less immediately combat useful than the sharingan. It'd mean some of Kakashi's tricks, like hiding under the Earth, would be totally useful, but other Kakashi would know that. It'd also mean that sharingan Kakashi has the edge in taijutsu, and even in a ninjutsu fight he'll know what byakugan Kakashi is going to do slightly before he does it (though the copy aspect is fortunately worthless between two Kakashi).
Byakugan Kakashi's best chance is somehow getting an ambush, attacks from blindspot, use clones, etc. but that's very hard even against a Kakashi without 360 vision. Still, multi-angle fighting isn't quite as easy if you can't see.
Terrain matters too. Best setup for Byakugan is a forest or similar with tons of hiding places, as he can hide (to an extent) by Sharingan can't. Best setup for Sharingan is open area.