First off, do NOT confuse two different issues: determining paternity and changing the child's name. Courts are very reticent to use a simple name change to circumvent a legitimate paternity process. Changing the name will not change the birth certificate or who is the legal father. Your "mail order" paternity test is also useless in court.

What you need to do is start a proper paternity action. This will likely take a DNA test that is performed in a legally acceptable way starting with someone neutral collecting the sample and sending it off. After the paternity is resolved in court, the court can also issue changes to the legal name and the birth certificate.

You're well advised to use a lawyer for all this. Both the current legal father and the father you are attempting to establish paternity for will obviously need notification.