The court having jurisdiction to determine the custody of the child would likely consider other factors more important than the alleged bigamy of the child's mother.
While the fact that your wife got married to you before her divorce was finalized might be considered in the child custody case, it probably will not get you the "leverage" it looks like you want. In fact, if you knew she was married and you married her anyway, that could backfire as a bad judgment call made by you as well.
Who gets custody of a child is mostly dependent on factors that affect parenting itself, not necessarily any historical bad act by one parent or the other. The courts are sophisticated enough to know that there are no "perfect" parents, and that everyone has done something they are not necessarily proud of. What they look at is who is the best person to take care of the child NOW. What are the child's needs and who is best suited to meet those. Just because a person made poor choices 5 years ago doesn't mean they are a bad parent today.
Most often, particularly when there are no serious issues about parenting, drug use, alcohol use, etc., the court will presume it is in every child's best interest to spend quality time with BOTH parents. Kids need both their mom and their dad, especially when mom and dad are healthy and a good caretaker/role-model.
I hope that helps you. Good luck, and take care of yourself and your kiddo.
As your wife was still married at the time of your wedding ceremony, then your marriage was 'void' as a matter of law, then you could indeed obtain an annulment. However, If, after you learned that your wife was not yet divorced and you nonetheless continued to live with her as man and wife until after such time as her prior divorce became final, then, under Texas law, your marriage has 'automatically' become valid and you would not qualify for an annulment.
If your wife honestly (albeit mistakenly) believed that her prior marriage had been legally terminated, then it is very unlikely that her conduct in "marrying" you would carry any weight in a child custody case. However, if she did indeed know that her prior marriage had not been dissolved then it might be a slightly different matter. The crime of Bigamy is rarely prosecuted and I assume that the commission of any such crime likely predated the birth of the child in issue. Child custody cases do not generally turn on misconduct that did not threaten harm to the child or directly undermine that party's ability to properly parent.
However, the credibility of the parties and their witnesses is almost always a significant issue in the trial of any case, including child custody cases. If your wife did indeed know that her prior marriage had not been dissolved, then it is very likely that she lied on your marriage license application. This could constitute perjury as well as the falsifying of a public record. Many Judges take such conduct very seriously, believing that if she was willing to lie on an important government form, she is likely to lie under oath in Court as well. This could help you in court, in that it would potentially serve to undermine and diminish the value and weight given to her testimony by the trial court.
Your marriage is NOT valid. I am not a licensed attorney in TX, however, rather than leverage which I think it will not give you, I think you should be concerned about moving on with your own life and the welfare of your child. Since the marriage is not valid, your paperwork and taxes should be filed accordingly. Depending on the age of the child she may try to get custody or visiting rights but this is another issue. The question here is whether you knew on marrying that your wife was already married. The other question is whether she was aware that the divorce was not finalized. In any case an annullment can be time consuming and difficult so you will want to consult an attorney.
Custody of the child is based upon the best interest of the child. Bigamy does not relate to the child. So no it would not help in a custody case. You can follow Texas procedure to have the marriage annulled based upon her bigamy.
From a Washington state attorney, I don't think that bigamy will get you nowhere in a support case. It's probably just a question of the parents' incomes.
Texas likes marriage. Even if your marriage was void when you entered it, most cases on this kind of situation find a way to confirm a valid marriage.
Generally, the Family Code in Texas favors joint custody. One of you will be Joint MANAGING Conservator, the other Joint POSSESSORY Conservator. The managing conservator will be the parent the child normally lives with, and the possessory conservator will be granted standard possession rights.
BIG factors in determining custody in Texas are 1) a history of family violence, and 2) any instance of child abuse. IF you could prove your wife defrauded you about the divorce, you might get some traction there, especially before a jury.
The fact that you are the primary parent in reality should get you a lot further than that bigamy deal, especially if you have people who know about your parenting of the child.
Regardless of your status as a married couple or just as unmarried parents of the child, the analysis would be pretty much the same.