fWAR assumes a pitcher has no control over his batted balls (FIP - fielding independent), which is wrong.
bWAR assumes a pitcher has complete control over his batted balls (RA/9 - runs allowed), which is even more wrong.
If I'm just looking up guys I will go with fWAR. I only use BRef for contract and service time info now, though sometimes I'll use their play index search.
This is why I have been leaning more towards xwOBA from Baseball Savant. This value is based on Ks, BBs, and the type of contact the pitcher allows including launch angle (grounders, liners, flyballs, pop ups) and exit velocity (some pitchers get hit harder than other pitchers). Since the outcomes for each batted ball type are averages from the entire league, park effects and defensive skill is removed.
The issue with xwOBA is that it's a rate stat, which doesn't reward a guy for pitching 200 innings more than a guy who only pitched 140. Combining xwOBA and innings is trivial, but it's still a manual process.
To me, that's what I want to know..."How good is the pitcher?" not, "How good were the ultimate results?".