Do I have to clean the barrel after shooting with BN coated bullets?

No, hBN isn't going foul like Cu or Pb does. The hBN in your barrel currently will wear out as you shoot it but to speed up the process you could always run a few patches through it so you start from a more consistent point.
 
Thanks for the reply! Guess I can take the Dewey along when I test these DTACs got a primer test to do with the Hornady HPBTs, too.

Yep, I do it all the time. I have a 6.5 Grendel that loves coated-107SMKs but hates anything 100/120/123gr coated, so I have to flip back and forth and do no barrel prep or at the most a few swipes with a patch/jag.