"BBQ Chicken" means so many things to so many people!
* True BBQ chicken, ie., half chickens stuck in a smoker pit like they have at BBQ joints around here, I'd say something malty but mild would go great, like a special bitter or brown ale. You'll notice at these places everything else is simple...the bread is plain white bread, plain cheap pickles....the meat and sauce are where all the complexity is, so in keeping with that I'd do a modestly understated brew.
* Grilled chicken slathered in red, sweet BBQ sauce...maybe something drier and less cloying like a Czech or Bavarian pils, just to get away from that heavy sweetness.
* Smoke roasted chicken (grilled indirect at higher temps with smoke, my favorite method), depending on how it is seasoned, could handle something a bit bolder I think. If its not bathed in KC Masterpiece (a bit of an unfortunate thing!), big IPA sounds like it'd be great. Actually I like the idea of a smoked bock with an herb crusted smoke roasted chicken.