Thanks for the chance to help. I am an attorney with over 12 years experience. Hopefully I can help you with your legal question.
Under TX law when a child is born during the marriage, the law presumes the husband to be the father.
This is a rebuttable presumption...the husband can file a motion with the court for a paternity test and if DNA evidence shows he is not the father, the court can order that the case.
So in your case, particularly with the birth certificate already in place, your spouse is also the father.
He can have a lawyer file a motion with the court to require testing to try and change this...it sounds like testing would show him not to be the father. But he has the burden at this point to ask the court to order the tests.
You can also take this action...you can have a lawyer ask the court to order the DNA test.
But unless you or he go to the court the birth certificate will continue to show your spouse as the father.
This is significant since as the father he has a legal right to custody of the child unless or until the court orders otherwise.
So you can agree on the terms of the divorce...but typically the court will assign joint custody
and will assign support (based on relative incomes)
It is possible to get the court to vary from this...to give custody fully to one parent or the other, if there is evidence to show this is best for the child. But custody is always subject to modification...so you may agree now and a year from now he could go back and seek more time with the child.
SO if your goal is to remove his ability to seek custody, the first step should be to ask the court for a DNA test to show he is not the father. I would do this at the same time as you file for divorce. If the test shows he is not the father the court will remove him from the birth certificate and will not award him custody.