If it were me, I would go after him for child support, I mean, after all, why should he have no responsibility in this just cause he "may want to spend more time" with your DD. Just because he may "want" to, does not mean the court would allow it. You've had your victory once. You put your faith in your HP, and he didn't steer you wrong. What I would do is pursue the child support, and if AH wanted to see my child, I'd say take it to court for visition. And I'd make sure in court I'd bring up that he's an alcoholic, and how he's not stable enough to take care of the child without supervision. Let the court decide. It may be "supervised" visitation, in which case, do you really think your AH would meet your D at a neutral place, supervised by Dept. of Social Services? I think not.
You say you don't really need child support, you can do it on your own. That is absolutely wonderful! But, what if circumstances, God forbid, changed for you? Your daughter has a right to be supported financially by her parent. You have that right to receive financial support from the other parent. It is part of his responsiblity. And if you don't need the money right now, put it in an account for your daughter, for college or for a house when she's in her 20's. JMO. I wish you all the best.