Marriage Advantages?

I'm in between a custody battle and was wondering does marriage have any advantages in family court? I know in some states they lower child support if you are married, however in my situation I don't want act child support or anything involve just want my LO. My lo father doesn't take care of her and hasn't seen her in months. I want full custody, I would like my fiancé to be her primary care giver basically her father. and when we get married I would like for her to have his last name, he wants to get married now and I'm afraid that the judge will think we got married for the wrong reasons.