I'm going to pose as a customer and put in a complaint about my coworker.
What did your coworker do for you to file a complaint?
It's the only way to deal with the problem I'm having with my coworker.
It's wrong to put in a fictitious complaint in about a coworker in order to get them in trouble. Speak with your manager and see if he or she can offer any insight into resolving the issue that has come up between you and your coworker.
The way I want to do it is less work.
It is less work, but it is also the worse option. If someone had a problem with you, you would want them to talk to a manager about it or talk to you about it. Do not go behind their back and make up a fake complaint.