Should allow my daughter to know her "part-time" father

I think you should tell your daughter the truth.Because one day she will recognize the difference to other kids around her, she will ask you about where her Dad is, why she doesn't see her father. You should tell her that she has father,and he loves her, but he can't stay with his daughter often because of some reasons. I mean don't until your daughter recognize the truth then you tell her. That will hurt her more seriously. The most important, you should let her know that she has a great mother who loves her so much.

If he doesn't want a relationship with her, then I would not push her into a relationship with him. My daughter knows her father and they occasionally talk, but my husband adopted her several years ago and he is her dad if you ask her. He is the one who has always cared even though he didn't have too!

Hmmm. this is quite a common case. I think (this is just my personal opinion) it best not to (yet). Because if you will be in a relationship it would be best that the new one will be looked up to as her father even if she may know later who really is her biological father. I guess a father is more of a relationship based on function, rather than relationship based on genetics. For me your ex is not her father, because a father will love and act on his love for His kid.