Pick the same character, fly back to the tower again, then go back to the character change screen. Instead what you should do is fly to the tower, then go to the character change screen. Once you have a two win streak, stop playing. I got lucky and had a streak going within four games of getting this quest, so it wasn’t so bad.
I have no guaranteed way to get this, other than maybe LFG-ing for a competent team. You need a two game streak for this work.