I have not been in this situation, but have you thought of consulting with a family law attorney? Usually the first consultation is free and you are under no obligation to sign a retainer agreement. They could probably do a background check on him and make sure he REALLY has turned his life around. You could get some information on what amount of child support you could expect to get in your situation, and maybe your answer would become more clear. Just based on my own experience (I was married to a drinker myself), I don't think he'd keep up payments if you just did an out-of-court agreement, and he'd probably only want to see her at his convenience if there was no court-ordered visitation schedule in place. I have a question, he seems eager to meet her, but do you think she wants to meet him and his family? Do you think it would disrupt her life if she had to go visit him on certain school breaks? Also, what do you hope to gain out of getting him to sign over paternal rights? Is this so your husband can adopt her?

Also (based on my experience being married to a drinker for many years), do you think he just wants to show off his new family to you and your daughter? I have no experience with people really turning their lives around for the better, but I do have experience with alcoholics being manipulative and needing constant attention.

Well I wasn't in that situation, but I do have children with someone who isn't my husband, but my husband has been there for my kiddos most of their lives. I was a single mom before my husband, even though I was technically married the father of one of my children. The father of my oldest child acted like he was going to want to be around but never was, and come to find out he had 2 other children around the same time I had my oldest. My youngest childs father I was married to and put his name on my oldest's birth certificate, and when we decided to have my youngest he acted like he was all for it. Well we were in the military, and he got stationed else where, and when I had our child he was all for her until he deployed. Then he started cheating on me, and we fell apart. When he got back she barely a year old, and he visited once, because we were getting divorced. After that he would call my mother acting like it was my fault that he never visited and that I kept the kids from him. He has called a couple of times in 10 years but said he would call back to talk to the kids if he could but wouldn't hear anything from him for a few years. I have never kept the kids from him, but he kept himself from the kids.
Anyways, my kids know my husband now as their dad, even though they know that he is not their biological father. They have asked if they look like their fathers but never ask why he never came around. My advice would be to talk to your daughter, which I know will be harder for you than it will be for her, but you need to know how she feels. Also, you should talk to him and see what his real intentions are. He needs to prove to you before you bring your daughter into it, because if he decides that he doesn't want anything to do with her, it will hurt her more. I grew up with out my father around and always thought he hated me, but as I got older I decided to reach out to him and we have a complicated relationship but it is something. If you choose to keep this from your daughter, in the future if she decides she wants to try and get some sort of relationship with him she might hold it against you. I know I do now as I have found out a lot of what happened for my dad to stop coming around. I had a mother who wouldn't allow him to call or visit, and he was not a violent man, my mom was just upset he had a new family.
Always put yourself in your daughter's shoes, how she would feel now, and how would she feel when she is your age if she chooses to build a relationship with him. I have always gone by the rule of answer questions that my kids have had about their father, and also helped them understand that their dad now loves them and has always been there for them and always will be no matter what. I make sure they know that in the future they will want to know more about their father, but until they can understand that it had nothing to do with them and that it was an issue between us, then you know they will better understand how to approach a relationship with her father.