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To a certain extent I think its usually not a problem for a barrister. Its all about the fair and even application of the law at the end of the day. My assumption is that its unusual to have a personal problem with a brief.
In my Dad's case, it was defending somebody alleged of a particularly brutal infanticide that was the end of the line for him. He knew he was doing what was right in the eyes of the law, but coming home to his young family posed some mental barriers that he wasn't prepared for.
Edit: ^ extreme example I know, but gives you the gyst.
Look up the "cab rank rule"
Summary here: https://en.wikipedia.org/wiki/Cab-rank_rule
Edit: I don't think many people know about this. You often see people on Twitter judging a barrister on who they have represented before as if there is any choice in the matter.
My Dad ended up quitting criminal law because of the emotional strain caused by putting huge amounts of effort into defending somebody who had comitted the most despicable crimes. He just wasn't cut out for it. Kept him awake at night. Probably still does.