Maybe I'm just desensitized but if it were me asking this question, I pretty much would not care to actually become associated with the kid.

I might also not like kids, or people in general, seeing as how I'm not even that great of a person and most people are barely okay, that what I lose by withholding the financial benefits is, quite honestly, not that significant.

Let the actual father and mother take care of her.

Only way I'd have a different position would be if An saw me as her mother, too, and not just her father's wife (as in, actually listened to me). Otherwise, she's almost a stranger and dealing with the situation becomes quite unnecessarily annoying abd burdensome for something I had no saying in.