I'm usually able to get past the artists's obnoxiousness when listening to their music. I primarily listen to classical music and jazz. Because most of the great artists in those genres are/were pretty much arrogant assholes, I don't really have much of a choice.
There's Charles Mingus, who habitually knocked people's teeth out when they didn't agree with him and threw his band members off stage when they didn't play well enough. Keith Jarrett, whose bandmates wouldn't even talk to him for 10 years and who regularly stops halfway through concerts and insults the audience when people cough. Miles Davis, who said a lot of very nasty things about white people, and regularly abused women, which he was quite proud of.
There's Richard Wagner whose personality and anti-semitic ideas have often been compared with those of Adolf Hitler, Bruckner who was a necrophiliac, Lully who was a rampant pedophile who hated Italians (even though he was himself Italian), Brahms whose favourite pastime was to sit at in front of an open window and attempt to kill neighbourhood cats with a bow and arrow, the misanthropic and arrogant Beethoven who caused his nephew to commit suicide, Mozart who rather paid for booze than food for his family etc.