I’ve personally had quite a hard time beating this mission and as I normally do whenever I find myself faced with hard situations, I look to cheese my way to victory. In my many attempts, I’ve found a method that is reliable.
1: To start out with, remove all the enemies in the area. I try to be a non-lethal stealth player, so I used stealth to knock out enemies and remove them from sight. There are two cameras located on the beams near the fighting pit so be sure to look out for those.
2: There is one lone guy in a room that is locked on either side. Aggress this npc by shooting at him through the window.
3: Run to the high ground immediately so the npc you just aggressed does not exit the room and attack you. If you kill him, it may alert the police. You can use any spot but I chose the rafters because the fighter did not approach me and just kept dodging side to side.
4: Shoot the fighter with any weapon. Once you down her, you will get credit for completing the mission. However, the fighter will be in a permanently downed state instead of getting back up and giving you some conversation bits about being able to use the gym and such. I have no idea how this will affect side mission progress in the future, so use this method at your own risk.