I might be reading your post wrong, but it seems to me your daughter doesn't want to cultivate a relationship with her biological dad..that it's your current boyfriend's idea that she should call her pops?
I say read your daughter's cues, not your SO's. From what you've shared with us about your ex, it sounds like he's emotionally unavailable, and your daughter is probably very aware of this, and really only makes contact with him to appease you.
I'd ask her what she wants to do. Maybe there's more to the story than you know. Maybe she's afraid of him because of his apparent alcoholism. Maybe she's resentful that he forces her to care for her half-sister. Maybe she's just fed up with his absentism and sorry half-attempts at love through late gifts. You don't say whether there's a court appointed visitation. Short of court a appointed visitation, why bother pursuing the relationship if it's one-sided? In the long-run it will only create undue pain and stress in your daughter's life. He's clearly not a suitable role model, and if he can't simply be there for her to simply show her fatherly love, what's the point?
Also, to address your concerns about creating ill-will between your child and ex due to your own feelings about him, from what I see here, there's no reason to fret about bad mouthing the guy, or being a road block to her having a relationship with her father. It sounds like he might be doing it for you through his own actions and words. I do agree that while he seems to be troubled, it's probably a good policy to not be seen lowering to his standards by cutting him down. Just be sure to communicate with your daughter, and don't be afraid to be honest. Just don't be mean as your being honest. Say, "Dad is dealing with a serious illness called alcoholism. It causes him to behave in a not-so-great way. I'm certain he doesn't mean to hurt your feelings" etc.
If at a later time in her life she decides to pursue a relationship with him, so be it. Just make sure it's on her terms, not anyone elses. Now if his behavior toward your daughter is downright dangerous or abusive, you have every right to discourage a relationship between them. In fact, it would be wise to go back to the courts for a reconsideration of visitation rights or even contact proper authorities for her safety's sake.