Paternity testing

Okay, so I am curious. I gave birth 11 weeks ago. 2 potential men could be the father. I am about 100% he’s not my now exboyfriends baby. Here’s the catch, I let him on the birth certificate because the other didn’t want anything to do with him. Long story short. I want to have him removed from the BC. Does that mean I have to have a paternity done on either of them? ....and if he turns out to not be the ex’s baby, do I have to have paternity done on the other and have him listed? Question is DO I HAVE TO?