This task is part of Arthur's storyline quests, and takes place in Maidenholm. The Club is only open after 2100 hours (9pm), so either sit and wait on the bench in the small front courtyard until then (seemed safe), or show up after 9pm. At that time, press the intercom button, show your invitation, and walk into the foyer. Talk to the Bobby, then enter the dressing room to the right. Find the costume and put it on. Talk to the Bobby again. If he is still not cooperative, go back to the dressing room and find another way in. Arthur will find the needed credentials on a small table in a hallway. Be sure to explore the club. No one will punish Arthur as long as he is in costume.
(Note: the text in the quest log states that the invitation is optional. Frankly, I can not remember how I got the invitation, but I do not think I did anything special to get it. Someone could post in the comments if there was any issue acquiring the invitation, or how Arthur can get into the club without one, and I will add that to the solution. Thanks.)
Update: Thanks to commenter nuttywray, who pointed out the invitation is on one of the benches right outside the club.