It sounds as though you are the wife in this scenario. It is legal for him to be on the birth certificate. And if he chooses to claim the child without genetic testing, there is not much you can do about it.
You should all be aware that he has a very limited window in which to dispute paternity and genetic testing would have to be a part of that.
This answer DOES NOT establish an attorney-client relationship. This answer is based on the limited information provided and is not intended to be conclusive advice. There are likely other factors that might influence or change the advice after a more lengthy consultation.
The man's wife has no paternity interests in this matter.
There is no reason for the man's wife to be involved in this matter.
If he decides to exercise his visitation rights then she will the step-parent of this child. As a step-parent she will have no legal rights regarding this child.
It is irrelevant that the woman knew that the man was married when they had the affair.
Sounds very harsh -- but that is Texas law.
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